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'___sec101'), - ('Using a deep neural network', 2, None, '___sec102'), - ('Wrapping it up', 2, None, '___sec103')]} + '___sec65')]} end of tocinfo --> @@ -229,7 +171,7 @@ MathJax.Hub.Config({ - Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks + Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning
@@ -303,44 +245,6 @@ MathJax.Hub.Config({
  • Which activation function should we use?
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  • Limitations of supervised learning with deep networks
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  • Convolutional Neural Networks (recognizing images)
    • -
  • Regular NNs don’t scale well to full images
    • -
  • 3D volumes of neurons
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  • Layers used to build CNNs
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  • Transforming images
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  • CNNs in brief
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  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
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  • Setting it up
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  • The MNIST dataset again
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  • Strong correlations
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  • Layers of a CNN
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  • Systematic reduction
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  • Importing Keras and Tensorflow
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  • Using TensorFlow backend
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  • Train the model
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  • Visualizing the results
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  • Running with Keras
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  • Final part
    • -
  • Final visualization
    • -
  • Fun links
    • -
  • Applications: solving ordinary differential equations with Neural Networks
    • -
  • Trial solution
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  • More details
    • -
  • Reformulating the problem
    • -
  • Estimating errors
    • -
  • Creating a simple Deep Neural Net
    • -
  • Setting up the code, feed forward part
    • -
  • Backpropagation
    • -
  • Gradient Descent
    • -
  • More on GD and cost function
    • -
  • An implementation of a Deep Neural Network
    • -
  • The final parts of the code
    • -
  • And adding Back propagation
    • -
  • Solving the ODE
    • -
  • Using neural network
    • -
  • Using a deep neural network
    • -
  • Wrapping it up
    • @@ -359,7 +263,7 @@ MathJax.Hub.Config({
    -

    Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks

    +

    Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning

    @@ -399,7 +303,7 @@ MathJax.Hub.Config({

  • 9
  • 10
  • ...
    • -
  • 105
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  • Which activation function should we use?
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  • Limitations of supervised learning with deep networks
    • -
  • Convolutional Neural Networks (recognizing images)
    • -
  • Regular NNs don’t scale well to full images
    • -
  • 3D volumes of neurons
    • -
  • Layers used to build CNNs
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  • Transforming images
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  • CNNs in brief
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  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
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  • Setting it up
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  • The MNIST dataset again
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  • Strong correlations
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  • Layers of a CNN
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  • Systematic reduction
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  • Prerequisites: Collect and pre-process data
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  • Importing Keras and Tensorflow
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  • Using TensorFlow backend
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  • Train the model
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  • Visualizing the results
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  • Running with Keras
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  • Trial solution
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  • Estimating errors
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  • Creating a simple Deep Neural Net
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  • Setting up the code, feed forward part
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  • Backpropagation
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  • Gradient Descent
    • -
  • More on GD and cost function
    • -
  • An implementation of a Deep Neural Network
    • -
  • The final parts of the code
    • -
  • And adding Back propagation
    • -
  • Solving the ODE
    • -
  • Using neural network
    • -
  • Using a deep neural network
    • -
  • Wrapping it up
    • @@ -384,7 +288,7 @@ a weight variable.
  • 10
  • 11
  • ...
    • -
  • 105
    • +
  • 67
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs002.html b/doc/pub/NeuralNet/html/._NeuralNet-bs002.html index d56670e32..2b508dde1 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs002.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs002.html @@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source - + -Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks +Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning @@ -141,65 +141,7 @@ Automatically generated HTML file from DocOnce source ('Limitations of supervised learning with deep networks', 2, None, - '___sec65'), - ('Convolutional Neural Networks (recognizing images)', - 2, - None, - '___sec66'), - ('Regular NNs don’t scale well to full images', - 2, - None, - '___sec67'), - ('3D volumes of neurons', 2, None, '___sec68'), - ('Layers used to build CNNs', 2, None, '___sec69'), - ('Transforming images', 2, None, '___sec70'), - ('CNNs in brief', 2, None, '___sec71'), - ('CNNs in more detail, building convolutional neural networks in ' - 'Tensorflow and Keras', - 2, - None, - '___sec72'), - ('Setting it up', 2, None, '___sec73'), - ('The MNIST dataset again', 2, None, '___sec74'), - ('Strong correlations', 2, None, '___sec75'), - ('Layers of a CNN', 2, None, '___sec76'), - ('Systematic reduction', 2, None, '___sec77'), - ('Prerequisites: Collect and pre-process data', - 2, - None, - '___sec78'), - ('Importing Keras and Tensorflow', 2, None, '___sec79'), - ('Using TensorFlow backend', 2, None, '___sec80'), - ('Train the model', 2, None, '___sec81'), - ('Visualizing the results', 2, None, '___sec82'), - ('Running with Keras', 2, None, '___sec83'), - ('Final part', 2, None, '___sec84'), - ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86'), - ('Applications: solving ordinary differential equations with ' - 'Neural Networks', - 2, - None, - '___sec87'), - ('Trial solution', 2, None, '___sec88'), - ('More details', 2, None, '___sec89'), - ('Reformulating the problem', 2, None, '___sec90'), - ('Estimating errors', 2, None, '___sec91'), - ('Creating a simple Deep Neural Net', 2, None, '___sec92'), - ('Setting up the code, feed forward part', 2, None, '___sec93'), - ('Backpropagation', 2, None, '___sec94'), - ('Gradient Descent', 2, None, '___sec95'), - ('More on GD and cost function', 2, None, '___sec96'), - ('An implementation of a Deep Neural Network', - 2, - None, - '___sec97'), - ('The final parts of the code', 2, None, '___sec98'), - ('And adding Back propagation', 2, None, '___sec99'), - ('Solving the ODE', 2, None, '___sec100'), - ('Using neural network', 2, None, '___sec101'), - ('Using a deep neural network', 2, None, '___sec102'), - ('Wrapping it up', 2, None, '___sec103')]} + '___sec65')]} end of tocinfo --> @@ -229,7 +171,7 @@ MathJax.Hub.Config({ - Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks + Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning
    @@ -303,44 +245,6 @@ MathJax.Hub.Config({
  • Which activation function should we use?
  • A top-down perspective on Neural networks
  • Limitations of supervised learning with deep networks
    • -
  • Convolutional Neural Networks (recognizing images)
    • -
  • Regular NNs don’t scale well to full images
    • -
  • 3D volumes of neurons
    • -
  • Layers used to build CNNs
    • -
  • Transforming images
    • -
  • CNNs in brief
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
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  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Using TensorFlow backend
    • -
  • Train the model
    • -
  • Visualizing the results
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • Fun links
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    • -
  • Trial solution
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  • More details
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  • Reformulating the problem
    • -
  • Estimating errors
    • -
  • Creating a simple Deep Neural Net
    • -
  • Setting up the code, feed forward part
    • -
  • Backpropagation
    • -
  • Gradient Descent
    • -
  • More on GD and cost function
    • -
  • An implementation of a Deep Neural Network
    • -
  • The final parts of the code
    • -
  • And adding Back propagation
    • -
  • Solving the ODE
    • -
  • Using neural network
    • -
  • Using a deep neural network
    • -
  • Wrapping it up
    • @@ -434,7 +338,7 @@ humanities to life science and medicine.
  • 11
  • 12
  • ...
    • -
  • 105
    • +
  • 67
  • »
    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs003.html b/doc/pub/NeuralNet/html/._NeuralNet-bs003.html index 0cb9b5c5c..dd188248b 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs003.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs003.html @@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source - + -Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks +Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning @@ -141,65 +141,7 @@ Automatically generated HTML file from DocOnce source ('Limitations of supervised learning with deep networks', 2, None, - '___sec65'), - ('Convolutional Neural Networks (recognizing images)', - 2, - None, - '___sec66'), - ('Regular NNs don’t scale well to full images', - 2, - None, - '___sec67'), - ('3D volumes of neurons', 2, None, '___sec68'), - ('Layers used to build CNNs', 2, None, '___sec69'), - ('Transforming images', 2, None, '___sec70'), - ('CNNs in brief', 2, None, '___sec71'), - ('CNNs in more detail, building convolutional neural networks in ' - 'Tensorflow and Keras', - 2, - None, - '___sec72'), - ('Setting it up', 2, None, '___sec73'), - ('The MNIST dataset again', 2, None, '___sec74'), - ('Strong correlations', 2, None, '___sec75'), - ('Layers of a CNN', 2, None, '___sec76'), - ('Systematic reduction', 2, None, '___sec77'), - ('Prerequisites: Collect and pre-process data', - 2, - None, - '___sec78'), - ('Importing Keras and Tensorflow', 2, None, '___sec79'), - ('Using TensorFlow backend', 2, None, '___sec80'), - ('Train the model', 2, None, '___sec81'), - ('Visualizing the results', 2, None, '___sec82'), - ('Running with Keras', 2, None, '___sec83'), - ('Final part', 2, None, '___sec84'), - ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86'), - ('Applications: solving ordinary differential equations with ' - 'Neural Networks', - 2, - None, - '___sec87'), - ('Trial solution', 2, None, '___sec88'), - ('More details', 2, None, '___sec89'), - ('Reformulating the problem', 2, None, '___sec90'), - ('Estimating errors', 2, None, '___sec91'), - ('Creating a simple Deep Neural Net', 2, None, '___sec92'), - ('Setting up the code, feed forward part', 2, None, '___sec93'), - ('Backpropagation', 2, None, '___sec94'), - ('Gradient Descent', 2, None, '___sec95'), - ('More on GD and cost function', 2, None, '___sec96'), - ('An implementation of a Deep Neural Network', - 2, - None, - '___sec97'), - ('The final parts of the code', 2, None, '___sec98'), - ('And adding Back propagation', 2, None, '___sec99'), - ('Solving the ODE', 2, None, '___sec100'), - ('Using neural network', 2, None, '___sec101'), - ('Using a deep neural network', 2, None, '___sec102'), - ('Wrapping it up', 2, None, '___sec103')]} + '___sec65')]} end of tocinfo --> @@ -229,7 +171,7 @@ MathJax.Hub.Config({ - Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks + Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning
    @@ -303,44 +245,6 @@ MathJax.Hub.Config({
  • Which activation function should we use?
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    • -
  • Convolutional Neural Networks (recognizing images)
    • -
  • Regular NNs don’t scale well to full images
    • -
  • 3D volumes of neurons
    • -
  • Layers used to build CNNs
    • -
  • Transforming images
    • -
  • CNNs in brief
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Using TensorFlow backend
    • -
  • Train the model
    • -
  • Visualizing the results
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • Fun links
    • -
  • Applications: solving ordinary differential equations with Neural Networks
    • -
  • Trial solution
    • -
  • More details
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  • Reformulating the problem
    • -
  • Estimating errors
    • -
  • Creating a simple Deep Neural Net
    • -
  • Setting up the code, feed forward part
    • -
  • Backpropagation
    • -
  • Gradient Descent
    • -
  • More on GD and cost function
    • -
  • An implementation of a Deep Neural Network
    • -
  • The final parts of the code
    • -
  • And adding Back propagation
    • -
  • Solving the ODE
    • -
  • Using neural network
    • -
  • Using a deep neural network
    • -
  • Wrapping it up
    • @@ -399,7 +303,7 @@ methods we discussed earlier.
  • 12
  • 13
  • ...
    • -
  • 105
    • +
  • 67
  • »
    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs004.html b/doc/pub/NeuralNet/html/._NeuralNet-bs004.html index 0de6da0a7..16cc942af 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs004.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs004.html @@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source - + -Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks +Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning @@ -141,65 +141,7 @@ Automatically generated HTML file from DocOnce source ('Limitations of supervised learning with deep networks', 2, None, - '___sec65'), - ('Convolutional Neural Networks (recognizing images)', - 2, - None, - '___sec66'), - ('Regular NNs don’t scale well to full images', - 2, - None, - '___sec67'), - ('3D volumes of neurons', 2, None, '___sec68'), - ('Layers used to build CNNs', 2, None, '___sec69'), - ('Transforming images', 2, None, '___sec70'), - ('CNNs in brief', 2, None, '___sec71'), - ('CNNs in more detail, building convolutional neural networks in ' - 'Tensorflow and Keras', - 2, - None, - '___sec72'), - ('Setting it up', 2, None, '___sec73'), - ('The MNIST dataset again', 2, None, '___sec74'), - ('Strong correlations', 2, None, '___sec75'), - ('Layers of a CNN', 2, None, '___sec76'), - ('Systematic reduction', 2, None, '___sec77'), - ('Prerequisites: Collect and pre-process data', - 2, - None, - '___sec78'), - ('Importing Keras and Tensorflow', 2, None, '___sec79'), - ('Using TensorFlow backend', 2, None, '___sec80'), - ('Train the model', 2, None, '___sec81'), - ('Visualizing the results', 2, None, '___sec82'), - ('Running with Keras', 2, None, '___sec83'), - ('Final part', 2, None, '___sec84'), - ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86'), - ('Applications: solving ordinary differential equations with ' - 'Neural Networks', - 2, - None, - '___sec87'), - ('Trial solution', 2, None, '___sec88'), - ('More details', 2, None, '___sec89'), - ('Reformulating the problem', 2, None, '___sec90'), - ('Estimating errors', 2, None, '___sec91'), - ('Creating a simple Deep Neural Net', 2, None, '___sec92'), - ('Setting up the code, feed forward part', 2, None, '___sec93'), - ('Backpropagation', 2, None, '___sec94'), - ('Gradient Descent', 2, None, '___sec95'), - ('More on GD and cost function', 2, None, '___sec96'), - ('An implementation of a Deep Neural Network', - 2, - None, - '___sec97'), - ('The final parts of the code', 2, None, '___sec98'), - ('And adding Back propagation', 2, None, '___sec99'), - ('Solving the ODE', 2, None, '___sec100'), - ('Using neural network', 2, None, '___sec101'), - ('Using a deep neural network', 2, None, '___sec102'), - ('Wrapping it up', 2, None, '___sec103')]} + '___sec65')]} end of tocinfo --> @@ -229,7 +171,7 @@ MathJax.Hub.Config({ - Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks + Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning
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  • Limitations of supervised learning with deep networks
    • -
  • Convolutional Neural Networks (recognizing images)
    • -
  • Regular NNs don’t scale well to full images
    • -
  • 3D volumes of neurons
    • -
  • Layers used to build CNNs
    • -
  • Transforming images
    • -
  • CNNs in brief
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Using TensorFlow backend
    • -
  • Train the model
    • -
  • Visualizing the results
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • Fun links
    • -
  • Applications: solving ordinary differential equations with Neural Networks
    • -
  • Trial solution
    • -
  • More details
    • -
  • Reformulating the problem
    • -
  • Estimating errors
    • -
  • Creating a simple Deep Neural Net
    • -
  • Setting up the code, feed forward part
    • -
  • Backpropagation
    • -
  • Gradient Descent
    • -
  • More on GD and cost function
    • -
  • An implementation of a Deep Neural Network
    • -
  • The final parts of the code
    • -
  • And adding Back propagation
    • -
  • Solving the ODE
    • -
  • Using neural network
    • -
  • Using a deep neural network
    • -
  • Wrapping it up
    • @@ -391,7 +295,7 @@ to all nodes in the subsequent layer, making this a so-called
  • 13
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  • ...
    • -
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  • »
    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs005.html b/doc/pub/NeuralNet/html/._NeuralNet-bs005.html index 2e3d0264a..aefb83526 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs005.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs005.html @@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source - + -Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks +Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning @@ -141,65 +141,7 @@ Automatically generated HTML file from DocOnce source ('Limitations of supervised learning with deep networks', 2, None, - '___sec65'), - ('Convolutional Neural Networks (recognizing images)', - 2, - None, - '___sec66'), - ('Regular NNs don’t scale well to full images', - 2, - None, - '___sec67'), - ('3D volumes of neurons', 2, None, '___sec68'), - ('Layers used to build CNNs', 2, None, '___sec69'), - ('Transforming images', 2, None, '___sec70'), - ('CNNs in brief', 2, None, '___sec71'), - ('CNNs in more detail, building convolutional neural networks in ' - 'Tensorflow and Keras', - 2, - None, - '___sec72'), - ('Setting it up', 2, None, '___sec73'), - ('The MNIST dataset again', 2, None, '___sec74'), - ('Strong correlations', 2, None, '___sec75'), - ('Layers of a CNN', 2, None, '___sec76'), - ('Systematic reduction', 2, None, '___sec77'), - ('Prerequisites: Collect and pre-process data', - 2, - None, - '___sec78'), - ('Importing Keras and Tensorflow', 2, None, '___sec79'), - ('Using TensorFlow backend', 2, None, '___sec80'), - ('Train the model', 2, None, '___sec81'), - ('Visualizing the results', 2, None, '___sec82'), - ('Running with Keras', 2, None, '___sec83'), - ('Final part', 2, None, '___sec84'), - ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86'), - ('Applications: solving ordinary differential equations with ' - 'Neural Networks', - 2, - None, - '___sec87'), - ('Trial solution', 2, None, '___sec88'), - ('More details', 2, None, '___sec89'), - ('Reformulating the problem', 2, None, '___sec90'), - ('Estimating errors', 2, None, '___sec91'), - ('Creating a simple Deep Neural Net', 2, None, '___sec92'), - ('Setting up the code, feed forward part', 2, None, '___sec93'), - ('Backpropagation', 2, None, '___sec94'), - ('Gradient Descent', 2, None, '___sec95'), - ('More on GD and cost function', 2, None, '___sec96'), - ('An implementation of a Deep Neural Network', - 2, - None, - '___sec97'), - ('The final parts of the code', 2, None, '___sec98'), - ('And adding Back propagation', 2, None, '___sec99'), - ('Solving the ODE', 2, None, '___sec100'), - ('Using neural network', 2, None, '___sec101'), - ('Using a deep neural network', 2, None, '___sec102'), - ('Wrapping it up', 2, None, '___sec103')]} + '___sec65')]} end of tocinfo --> @@ -229,7 +171,7 @@ MathJax.Hub.Config({ - Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks + Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning
    @@ -303,44 +245,6 @@ MathJax.Hub.Config({
  • Which activation function should we use?
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  • Limitations of supervised learning with deep networks
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  • Convolutional Neural Networks (recognizing images)
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  • Regular NNs don’t scale well to full images
    • -
  • 3D volumes of neurons
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  • Transforming images
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  • The MNIST dataset again
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  • Using TensorFlow backend
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  • Visualizing the results
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  • Running with Keras
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  • Final part
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  • Final visualization
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  • Fun links
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  • Applications: solving ordinary differential equations with Neural Networks
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  • Trial solution
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  • More details
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  • Reformulating the problem
    • -
  • Estimating errors
    • -
  • Creating a simple Deep Neural Net
    • -
  • Setting up the code, feed forward part
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  • Backpropagation
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  • Gradient Descent
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  • More on GD and cost function
    • -
  • An implementation of a Deep Neural Network
    • -
  • The final parts of the code
    • -
  • And adding Back propagation
    • -
  • Solving the ODE
    • -
  • Using neural network
    • -
  • Using a deep neural network
    • -
  • Wrapping it up
    • @@ -400,7 +304,7 @@ recognition.
  • 14
  • 15
  • ...
    • -
  • 105
    • +
  • 67
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs006.html b/doc/pub/NeuralNet/html/._NeuralNet-bs006.html index d0da32e46..7903ddbff 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs006.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs006.html @@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source - + -Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks +Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning @@ -141,65 +141,7 @@ Automatically generated HTML file from DocOnce source ('Limitations of supervised learning with deep networks', 2, None, - '___sec65'), - ('Convolutional Neural Networks (recognizing images)', - 2, - None, - '___sec66'), - ('Regular NNs don’t scale well to full images', - 2, - None, - '___sec67'), - ('3D volumes of neurons', 2, None, '___sec68'), - ('Layers used to build CNNs', 2, None, '___sec69'), - ('Transforming images', 2, None, '___sec70'), - ('CNNs in brief', 2, None, '___sec71'), - ('CNNs in more detail, building convolutional neural networks in ' - 'Tensorflow and Keras', - 2, - None, - '___sec72'), - ('Setting it up', 2, None, '___sec73'), - ('The MNIST dataset again', 2, None, '___sec74'), - ('Strong correlations', 2, None, '___sec75'), - ('Layers of a CNN', 2, None, '___sec76'), - ('Systematic reduction', 2, None, '___sec77'), - ('Prerequisites: Collect and pre-process data', - 2, - None, - '___sec78'), - ('Importing Keras and Tensorflow', 2, None, '___sec79'), - ('Using TensorFlow backend', 2, None, '___sec80'), - ('Train the model', 2, None, '___sec81'), - ('Visualizing the results', 2, None, '___sec82'), - ('Running with Keras', 2, None, '___sec83'), - ('Final part', 2, None, '___sec84'), - ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86'), - ('Applications: solving ordinary differential equations with ' - 'Neural Networks', - 2, - None, - '___sec87'), - ('Trial solution', 2, None, '___sec88'), - ('More details', 2, None, '___sec89'), - ('Reformulating the problem', 2, None, '___sec90'), - ('Estimating errors', 2, None, '___sec91'), - ('Creating a simple Deep Neural Net', 2, None, '___sec92'), - ('Setting up the code, feed forward part', 2, None, '___sec93'), - ('Backpropagation', 2, None, '___sec94'), - ('Gradient Descent', 2, None, '___sec95'), - ('More on GD and cost function', 2, None, '___sec96'), - ('An implementation of a Deep Neural Network', - 2, - None, - '___sec97'), - ('The final parts of the code', 2, None, '___sec98'), - ('And adding Back propagation', 2, None, '___sec99'), - ('Solving the ODE', 2, None, '___sec100'), - ('Using neural network', 2, None, '___sec101'), - ('Using a deep neural network', 2, None, '___sec102'), - ('Wrapping it up', 2, None, '___sec103')]} + '___sec65')]} end of tocinfo --> @@ -229,7 +171,7 @@ MathJax.Hub.Config({ - Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks + Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning
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  • Which activation function should we use?
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  • Regular NNs don’t scale well to full images
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  • 3D volumes of neurons
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  • Transforming images
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  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
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  • The MNIST dataset again
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  • Strong correlations
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  • Layers of a CNN
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  • Importing Keras and Tensorflow
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  • Using TensorFlow backend
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  • Running with Keras
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  • Trial solution
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  • Creating a simple Deep Neural Net
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  • Gradient Descent
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  • More on GD and cost function
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  • An implementation of a Deep Neural Network
    • -
  • The final parts of the code
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  • And adding Back propagation
    • -
  • Solving the ODE
    • -
  • Using neural network
    • -
  • Using a deep neural network
    • -
  • Wrapping it up
    • @@ -392,7 +296,7 @@ especially well-suited for handwriting and speech recognition.
  • 15
  • 16
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    • -
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs007.html b/doc/pub/NeuralNet/html/._NeuralNet-bs007.html index b35c2a78b..e99420ef8 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs007.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs007.html @@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source - + -Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks +Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning @@ -141,65 +141,7 @@ Automatically generated HTML file from DocOnce source ('Limitations of supervised learning with deep networks', 2, None, - '___sec65'), - ('Convolutional Neural Networks (recognizing images)', - 2, - None, - '___sec66'), - ('Regular NNs don’t scale well to full images', - 2, - None, - '___sec67'), - ('3D volumes of neurons', 2, None, '___sec68'), - ('Layers used to build CNNs', 2, None, '___sec69'), - ('Transforming images', 2, None, '___sec70'), - ('CNNs in brief', 2, None, '___sec71'), - ('CNNs in more detail, building convolutional neural networks in ' - 'Tensorflow and Keras', - 2, - None, - '___sec72'), - ('Setting it up', 2, None, '___sec73'), - ('The MNIST dataset again', 2, None, '___sec74'), - ('Strong correlations', 2, None, '___sec75'), - ('Layers of a CNN', 2, None, '___sec76'), - ('Systematic reduction', 2, None, '___sec77'), - ('Prerequisites: Collect and pre-process data', - 2, - None, - '___sec78'), - ('Importing Keras and Tensorflow', 2, None, '___sec79'), - ('Using TensorFlow backend', 2, None, '___sec80'), - ('Train the model', 2, None, '___sec81'), - ('Visualizing the results', 2, None, '___sec82'), - ('Running with Keras', 2, None, '___sec83'), - ('Final part', 2, None, '___sec84'), - ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86'), - ('Applications: solving ordinary differential equations with ' - 'Neural Networks', - 2, - None, - '___sec87'), - ('Trial solution', 2, None, '___sec88'), - ('More details', 2, None, '___sec89'), - ('Reformulating the problem', 2, None, '___sec90'), - ('Estimating errors', 2, None, '___sec91'), - ('Creating a simple Deep Neural Net', 2, None, '___sec92'), - ('Setting up the code, feed forward part', 2, None, '___sec93'), - ('Backpropagation', 2, None, '___sec94'), - ('Gradient Descent', 2, None, '___sec95'), - ('More on GD and cost function', 2, None, '___sec96'), - ('An implementation of a Deep Neural Network', - 2, - None, - '___sec97'), - ('The final parts of the code', 2, None, '___sec98'), - ('And adding Back propagation', 2, None, '___sec99'), - ('Solving the ODE', 2, None, '___sec100'), - ('Using neural network', 2, None, '___sec101'), - ('Using a deep neural network', 2, None, '___sec102'), - ('Wrapping it up', 2, None, '___sec103')]} + '___sec65')]} end of tocinfo --> @@ -229,7 +171,7 @@ MathJax.Hub.Config({ - Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks + Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning
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  • Which activation function should we use?
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  • Regular NNs don’t scale well to full images
    • -
  • 3D volumes of neurons
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  • Layers used to build CNNs
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  • Transforming images
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  • More on GD and cost function
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  • An implementation of a Deep Neural Network
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  • The final parts of the code
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    • -
  • Solving the ODE
    • -
  • Using neural network
    • -
  • Using a deep neural network
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  • Wrapping it up
    • @@ -393,7 +297,7 @@ type of NN due the unusual activation functions.
  • 16
  • 17
  • ...
    • -
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    • +
  • 67
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs008.html b/doc/pub/NeuralNet/html/._NeuralNet-bs008.html index 819546894..38644dd1b 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs008.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs008.html @@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source - + -Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks +Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning @@ -141,65 +141,7 @@ Automatically generated HTML file from DocOnce source ('Limitations of supervised learning with deep networks', 2, None, - '___sec65'), - ('Convolutional Neural Networks (recognizing images)', - 2, - None, - '___sec66'), - ('Regular NNs don’t scale well to full images', - 2, - None, - '___sec67'), - ('3D volumes of neurons', 2, None, '___sec68'), - ('Layers used to build CNNs', 2, None, '___sec69'), - ('Transforming images', 2, None, '___sec70'), - ('CNNs in brief', 2, None, '___sec71'), - ('CNNs in more detail, building convolutional neural networks in ' - 'Tensorflow and Keras', - 2, - None, - '___sec72'), - ('Setting it up', 2, None, '___sec73'), - ('The MNIST dataset again', 2, None, '___sec74'), - ('Strong correlations', 2, None, '___sec75'), - ('Layers of a CNN', 2, None, '___sec76'), - ('Systematic reduction', 2, None, '___sec77'), - ('Prerequisites: Collect and pre-process data', - 2, - None, - '___sec78'), - ('Importing Keras and Tensorflow', 2, None, '___sec79'), - ('Using TensorFlow backend', 2, None, '___sec80'), - ('Train the model', 2, None, '___sec81'), - ('Visualizing the results', 2, None, '___sec82'), - ('Running with Keras', 2, None, '___sec83'), - ('Final part', 2, None, '___sec84'), - ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86'), - ('Applications: solving ordinary differential equations with ' - 'Neural Networks', - 2, - None, - '___sec87'), - ('Trial solution', 2, None, '___sec88'), - ('More details', 2, None, '___sec89'), - ('Reformulating the problem', 2, None, '___sec90'), - ('Estimating errors', 2, None, '___sec91'), - ('Creating a simple Deep Neural Net', 2, None, '___sec92'), - ('Setting up the code, feed forward part', 2, None, '___sec93'), - ('Backpropagation', 2, None, '___sec94'), - ('Gradient Descent', 2, None, '___sec95'), - ('More on GD and cost function', 2, None, '___sec96'), - ('An implementation of a Deep Neural Network', - 2, - None, - '___sec97'), - ('The final parts of the code', 2, None, '___sec98'), - ('And adding Back propagation', 2, None, '___sec99'), - ('Solving the ODE', 2, None, '___sec100'), - ('Using neural network', 2, None, '___sec101'), - ('Using a deep neural network', 2, None, '___sec102'), - ('Wrapping it up', 2, None, '___sec103')]} + '___sec65')]} end of tocinfo --> @@ -229,7 +171,7 @@ MathJax.Hub.Config({ - Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks + Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning
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  • Which activation function should we use?
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  • Regular NNs don’t scale well to full images
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  • 3D volumes of neurons
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  • Transforming images
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  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
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  • Strong correlations
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  • Gradient Descent
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  • More on GD and cost function
    • -
  • An implementation of a Deep Neural Network
    • -
  • The final parts of the code
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  • And adding Back propagation
    • -
  • Solving the ODE
    • -
  • Using neural network
    • -
  • Using a deep neural network
    • -
  • Wrapping it up
    • @@ -391,7 +295,7 @@ Such networks are often called multilayer perceptrons (MLPs).
  • 17
  • 18
  • ...
    • -
  • 105
    • +
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  • »
    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs009.html b/doc/pub/NeuralNet/html/._NeuralNet-bs009.html index 646f7017b..cd08f2b71 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs009.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs009.html @@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source - + -Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks +Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning @@ -141,65 +141,7 @@ Automatically generated HTML file from DocOnce source ('Limitations of supervised learning with deep networks', 2, None, - '___sec65'), - ('Convolutional Neural Networks (recognizing images)', - 2, - None, - '___sec66'), - ('Regular NNs don’t scale well to full images', - 2, - None, - '___sec67'), - ('3D volumes of neurons', 2, None, '___sec68'), - ('Layers used to build CNNs', 2, None, '___sec69'), - ('Transforming images', 2, None, '___sec70'), - ('CNNs in brief', 2, None, '___sec71'), - ('CNNs in more detail, building convolutional neural networks in ' - 'Tensorflow and Keras', - 2, - None, - '___sec72'), - ('Setting it up', 2, None, '___sec73'), - ('The MNIST dataset again', 2, None, '___sec74'), - ('Strong correlations', 2, None, '___sec75'), - ('Layers of a CNN', 2, None, '___sec76'), - ('Systematic reduction', 2, None, '___sec77'), - ('Prerequisites: Collect and pre-process data', - 2, - None, - '___sec78'), - ('Importing Keras and Tensorflow', 2, None, '___sec79'), - ('Using TensorFlow backend', 2, None, '___sec80'), - ('Train the model', 2, None, '___sec81'), - ('Visualizing the results', 2, None, '___sec82'), - ('Running with Keras', 2, None, '___sec83'), - ('Final part', 2, None, '___sec84'), - ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86'), - ('Applications: solving ordinary differential equations with ' - 'Neural Networks', - 2, - None, - '___sec87'), - ('Trial solution', 2, None, '___sec88'), - ('More details', 2, None, '___sec89'), - ('Reformulating the problem', 2, None, '___sec90'), - ('Estimating errors', 2, None, '___sec91'), - ('Creating a simple Deep Neural Net', 2, None, '___sec92'), - ('Setting up the code, feed forward part', 2, None, '___sec93'), - ('Backpropagation', 2, None, '___sec94'), - ('Gradient Descent', 2, None, '___sec95'), - ('More on GD and cost function', 2, None, '___sec96'), - ('An implementation of a Deep Neural Network', - 2, - None, - '___sec97'), - ('The final parts of the code', 2, None, '___sec98'), - ('And adding Back propagation', 2, None, '___sec99'), - ('Solving the ODE', 2, None, '___sec100'), - ('Using neural network', 2, None, '___sec101'), - ('Using a deep neural network', 2, None, '___sec102'), - ('Wrapping it up', 2, None, '___sec103')]} + '___sec65')]} end of tocinfo --> @@ -229,7 +171,7 @@ MathJax.Hub.Config({ - Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks + Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning
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  • Regular NNs don’t scale well to full images
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  • 3D volumes of neurons
    • -
  • Layers used to build CNNs
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  • Transforming images
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  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
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  • Setting it up
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  • The MNIST dataset again
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  • Strong correlations
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  • Layers of a CNN
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    • -
  • Final visualization
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  • Fun links
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  • Applications: solving ordinary differential equations with Neural Networks
    • -
  • Trial solution
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  • More details
    • -
  • Reformulating the problem
    • -
  • Estimating errors
    • -
  • Creating a simple Deep Neural Net
    • -
  • Setting up the code, feed forward part
    • -
  • Backpropagation
    • -
  • Gradient Descent
    • -
  • More on GD and cost function
    • -
  • An implementation of a Deep Neural Network
    • -
  • The final parts of the code
    • -
  • And adding Back propagation
    • -
  • Solving the ODE
    • -
  • Using neural network
    • -
  • Using a deep neural network
    • -
  • Wrapping it up
    • @@ -396,7 +300,7 @@ as to not restrict the range of output values.
  • 18
  • 19
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    • -
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  • Creating a simple Deep Neural Net
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  • Setting up the code, feed forward part
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  • Gradient Descent
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  • More on GD and cost function
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  • An implementation of a Deep Neural Network
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  • The final parts of the code
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  • And adding Back propagation
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  • Solving the ODE
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  • Using neural network
    • -
  • Using a deep neural network
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  • Wrapping it up
    • @@ -397,7 +301,7 @@ of the outputs of all neurons in the previous layer.
  • 19
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    • -
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  • More on GD and cost function
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  • An implementation of a Deep Neural Network
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  • The final parts of the code
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  • And adding Back propagation
    • -
  • Solving the ODE
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  • Using neural network
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  • Using a deep neural network
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  • Wrapping it up
    • @@ -426,7 +330,7 @@ is obtained.
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  • More on GD and cost function
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  • An implementation of a Deep Neural Network
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  • Solving the ODE
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  • Wrapping it up
    • @@ -408,7 +312,7 @@ $$
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  • Gradient Descent
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  • More on GD and cost function
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  • An implementation of a Deep Neural Network
    • -
  • The final parts of the code
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  • And adding Back propagation
    • -
  • Solving the ODE
    • -
  • Using neural network
    • -
  • Using a deep neural network
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  • Wrapping it up
    • @@ -399,7 +303,7 @@ variables are the input values \( x_n \).
  • 22
  • 23
  • ...
    • -
  • 105
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  • 67
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs014.html b/doc/pub/NeuralNet/html/._NeuralNet-bs014.html index bca27d454..ebbdaaf35 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs014.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs014.html @@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source - + -Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks +Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning @@ -141,65 +141,7 @@ Automatically generated HTML file from DocOnce source ('Limitations of supervised learning with deep networks', 2, None, - '___sec65'), - ('Convolutional Neural Networks (recognizing images)', - 2, - None, - '___sec66'), - ('Regular NNs don’t scale well to full images', - 2, - None, - '___sec67'), - ('3D volumes of neurons', 2, None, '___sec68'), - ('Layers used to build CNNs', 2, None, '___sec69'), - ('Transforming images', 2, None, '___sec70'), - ('CNNs in brief', 2, None, '___sec71'), - ('CNNs in more detail, building convolutional neural networks in ' - 'Tensorflow and Keras', - 2, - None, - '___sec72'), - ('Setting it up', 2, None, '___sec73'), - ('The MNIST dataset again', 2, None, '___sec74'), - ('Strong correlations', 2, None, '___sec75'), - ('Layers of a CNN', 2, None, '___sec76'), - ('Systematic reduction', 2, None, '___sec77'), - ('Prerequisites: Collect and pre-process data', - 2, - None, - '___sec78'), - ('Importing Keras and Tensorflow', 2, None, '___sec79'), - ('Using TensorFlow backend', 2, None, '___sec80'), - ('Train the model', 2, None, '___sec81'), - ('Visualizing the results', 2, None, '___sec82'), - ('Running with Keras', 2, None, '___sec83'), - ('Final part', 2, None, '___sec84'), - ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86'), - ('Applications: solving ordinary differential equations with ' - 'Neural Networks', - 2, - None, - '___sec87'), - ('Trial solution', 2, None, '___sec88'), - ('More details', 2, None, '___sec89'), - ('Reformulating the problem', 2, None, '___sec90'), - ('Estimating errors', 2, None, '___sec91'), - ('Creating a simple Deep Neural Net', 2, None, '___sec92'), - ('Setting up the code, feed forward part', 2, None, '___sec93'), - ('Backpropagation', 2, None, '___sec94'), - ('Gradient Descent', 2, None, '___sec95'), - ('More on GD and cost function', 2, None, '___sec96'), - ('An implementation of a Deep Neural Network', - 2, - None, - '___sec97'), - ('The final parts of the code', 2, None, '___sec98'), - ('And adding Back propagation', 2, None, '___sec99'), - ('Solving the ODE', 2, None, '___sec100'), - ('Using neural network', 2, None, '___sec101'), - ('Using a deep neural network', 2, None, '___sec102'), - ('Wrapping it up', 2, None, '___sec103')]} + '___sec65')]} end of tocinfo --> @@ -229,7 +171,7 @@ MathJax.Hub.Config({ - Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks + Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning
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  • 3D volumes of neurons
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  • Layers used to build CNNs
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  • Transforming images
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  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
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  • The MNIST dataset again
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  • Strong correlations
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    • -
  • Creating a simple Deep Neural Net
    • -
  • Setting up the code, feed forward part
    • -
  • Backpropagation
    • -
  • Gradient Descent
    • -
  • More on GD and cost function
    • -
  • An implementation of a Deep Neural Network
    • -
  • The final parts of the code
    • -
  • And adding Back propagation
    • -
  • Solving the ODE
    • -
  • Using neural network
    • -
  • Using a deep neural network
    • -
  • Wrapping it up
    • @@ -408,7 +312,7 @@ flexibility of a neural network.
  • 23
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  • ...
    • -
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    • +
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  • »
    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs015.html b/doc/pub/NeuralNet/html/._NeuralNet-bs015.html index a77a7e7ad..b6e93efe1 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs015.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs015.html @@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source - + -Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks +Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning @@ -141,65 +141,7 @@ Automatically generated HTML file from DocOnce source ('Limitations of supervised learning with deep networks', 2, None, - '___sec65'), - ('Convolutional Neural Networks (recognizing images)', - 2, - None, - '___sec66'), - ('Regular NNs don’t scale well to full images', - 2, - None, - '___sec67'), - ('3D volumes of neurons', 2, None, '___sec68'), - ('Layers used to build CNNs', 2, None, '___sec69'), - ('Transforming images', 2, None, '___sec70'), - ('CNNs in brief', 2, None, '___sec71'), - ('CNNs in more detail, building convolutional neural networks in ' - 'Tensorflow and Keras', - 2, - None, - '___sec72'), - ('Setting it up', 2, None, '___sec73'), - ('The MNIST dataset again', 2, None, '___sec74'), - ('Strong correlations', 2, None, '___sec75'), - ('Layers of a CNN', 2, None, '___sec76'), - ('Systematic reduction', 2, None, '___sec77'), - ('Prerequisites: Collect and pre-process data', - 2, - None, - '___sec78'), - ('Importing Keras and Tensorflow', 2, None, '___sec79'), - ('Using TensorFlow backend', 2, None, '___sec80'), - ('Train the model', 2, None, '___sec81'), - ('Visualizing the results', 2, None, '___sec82'), - ('Running with Keras', 2, None, '___sec83'), - ('Final part', 2, None, '___sec84'), - ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86'), - ('Applications: solving ordinary differential equations with ' - 'Neural Networks', - 2, - None, - '___sec87'), - ('Trial solution', 2, None, '___sec88'), - ('More details', 2, None, '___sec89'), - ('Reformulating the problem', 2, None, '___sec90'), - ('Estimating errors', 2, None, '___sec91'), - ('Creating a simple Deep Neural Net', 2, None, '___sec92'), - ('Setting up the code, feed forward part', 2, None, '___sec93'), - ('Backpropagation', 2, None, '___sec94'), - ('Gradient Descent', 2, None, '___sec95'), - ('More on GD and cost function', 2, None, '___sec96'), - ('An implementation of a Deep Neural Network', - 2, - None, - '___sec97'), - ('The final parts of the code', 2, None, '___sec98'), - ('And adding Back propagation', 2, None, '___sec99'), - ('Solving the ODE', 2, None, '___sec100'), - ('Using neural network', 2, None, '___sec101'), - ('Using a deep neural network', 2, None, '___sec102'), - ('Wrapping it up', 2, None, '___sec103')]} + '___sec65')]} end of tocinfo --> @@ -229,7 +171,7 @@ MathJax.Hub.Config({ - Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks + Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning
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  • Which activation function should we use?
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  • Regular NNs don’t scale well to full images
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  • 3D volumes of neurons
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  • Running with Keras
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  • Fun links
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  • Trial solution
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  • Reformulating the problem
    • -
  • Estimating errors
    • -
  • Creating a simple Deep Neural Net
    • -
  • Setting up the code, feed forward part
    • -
  • Backpropagation
    • -
  • Gradient Descent
    • -
  • More on GD and cost function
    • -
  • An implementation of a Deep Neural Network
    • -
  • The final parts of the code
    • -
  • And adding Back propagation
    • -
  • Solving the ODE
    • -
  • Using neural network
    • -
  • Using a deep neural network
    • -
  • Wrapping it up
    • @@ -418,7 +322,7 @@ $$
  • 24
  • 25
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    • -
  • 105
    • +
  • 67
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs016.html b/doc/pub/NeuralNet/html/._NeuralNet-bs016.html index 709546317..4ff8baa98 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs016.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs016.html @@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source - + -Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks +Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning @@ -141,65 +141,7 @@ Automatically generated HTML file from DocOnce source ('Limitations of supervised learning with deep networks', 2, None, - '___sec65'), - ('Convolutional Neural Networks (recognizing images)', - 2, - None, - '___sec66'), - ('Regular NNs don’t scale well to full images', - 2, - None, - '___sec67'), - ('3D volumes of neurons', 2, None, '___sec68'), - ('Layers used to build CNNs', 2, None, '___sec69'), - ('Transforming images', 2, None, '___sec70'), - ('CNNs in brief', 2, None, '___sec71'), - ('CNNs in more detail, building convolutional neural networks in ' - 'Tensorflow and Keras', - 2, - None, - '___sec72'), - ('Setting it up', 2, None, '___sec73'), - ('The MNIST dataset again', 2, None, '___sec74'), - ('Strong correlations', 2, None, '___sec75'), - ('Layers of a CNN', 2, None, '___sec76'), - ('Systematic reduction', 2, None, '___sec77'), - ('Prerequisites: Collect and pre-process data', - 2, - None, - '___sec78'), - ('Importing Keras and Tensorflow', 2, None, '___sec79'), - ('Using TensorFlow backend', 2, None, '___sec80'), - ('Train the model', 2, None, '___sec81'), - ('Visualizing the results', 2, None, '___sec82'), - ('Running with Keras', 2, None, '___sec83'), - ('Final part', 2, None, '___sec84'), - ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86'), - ('Applications: solving ordinary differential equations with ' - 'Neural Networks', - 2, - None, - '___sec87'), - ('Trial solution', 2, None, '___sec88'), - ('More details', 2, None, '___sec89'), - ('Reformulating the problem', 2, None, '___sec90'), - ('Estimating errors', 2, None, '___sec91'), - ('Creating a simple Deep Neural Net', 2, None, '___sec92'), - ('Setting up the code, feed forward part', 2, None, '___sec93'), - ('Backpropagation', 2, None, '___sec94'), - ('Gradient Descent', 2, None, '___sec95'), - ('More on GD and cost function', 2, None, '___sec96'), - ('An implementation of a Deep Neural Network', - 2, - None, - '___sec97'), - ('The final parts of the code', 2, None, '___sec98'), - ('And adding Back propagation', 2, None, '___sec99'), - ('Solving the ODE', 2, None, '___sec100'), - ('Using neural network', 2, None, '___sec101'), - ('Using a deep neural network', 2, None, '___sec102'), - ('Wrapping it up', 2, None, '___sec103')]} + '___sec65')]} end of tocinfo --> @@ -229,7 +171,7 @@ MathJax.Hub.Config({ - Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks + Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning
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  • Regular NNs don’t scale well to full images
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  • 3D volumes of neurons
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  • Gradient Descent
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  • More on GD and cost function
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  • An implementation of a Deep Neural Network
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  • The final parts of the code
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  • And adding Back propagation
    • -
  • Solving the ODE
    • -
  • Using neural network
    • -
  • Using a deep neural network
    • -
  • Wrapping it up
    • @@ -402,7 +306,7 @@ used as input to the activation functions. For each operation
  • 25
  • 26
  • ...
    • -
  • 105
    • +
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs017.html b/doc/pub/NeuralNet/html/._NeuralNet-bs017.html index a62a3d261..f8157f689 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs017.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs017.html @@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source - + -Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks +Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning @@ -141,65 +141,7 @@ Automatically generated HTML file from DocOnce source ('Limitations of supervised learning with deep networks', 2, None, - '___sec65'), - ('Convolutional Neural Networks (recognizing images)', - 2, - None, - '___sec66'), - ('Regular NNs don’t scale well to full images', - 2, - None, - '___sec67'), - ('3D volumes of neurons', 2, None, '___sec68'), - ('Layers used to build CNNs', 2, None, '___sec69'), - ('Transforming images', 2, None, '___sec70'), - ('CNNs in brief', 2, None, '___sec71'), - ('CNNs in more detail, building convolutional neural networks in ' - 'Tensorflow and Keras', - 2, - None, - '___sec72'), - ('Setting it up', 2, None, '___sec73'), - ('The MNIST dataset again', 2, None, '___sec74'), - ('Strong correlations', 2, None, '___sec75'), - ('Layers of a CNN', 2, None, '___sec76'), - ('Systematic reduction', 2, None, '___sec77'), - ('Prerequisites: Collect and pre-process data', - 2, - None, - '___sec78'), - ('Importing Keras and Tensorflow', 2, None, '___sec79'), - ('Using TensorFlow backend', 2, None, '___sec80'), - ('Train the model', 2, None, '___sec81'), - ('Visualizing the results', 2, None, '___sec82'), - ('Running with Keras', 2, None, '___sec83'), - ('Final part', 2, None, '___sec84'), - ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86'), - ('Applications: solving ordinary differential equations with ' - 'Neural Networks', - 2, - None, - '___sec87'), - ('Trial solution', 2, None, '___sec88'), - ('More details', 2, None, '___sec89'), - ('Reformulating the problem', 2, None, '___sec90'), - ('Estimating errors', 2, None, '___sec91'), - ('Creating a simple Deep Neural Net', 2, None, '___sec92'), - ('Setting up the code, feed forward part', 2, None, '___sec93'), - ('Backpropagation', 2, None, '___sec94'), - ('Gradient Descent', 2, None, '___sec95'), - ('More on GD and cost function', 2, None, '___sec96'), - ('An implementation of a Deep Neural Network', - 2, - None, - '___sec97'), - ('The final parts of the code', 2, None, '___sec98'), - ('And adding Back propagation', 2, None, '___sec99'), - ('Solving the ODE', 2, None, '___sec100'), - ('Using neural network', 2, None, '___sec101'), - ('Using a deep neural network', 2, None, '___sec102'), - ('Wrapping it up', 2, None, '___sec103')]} + '___sec65')]} end of tocinfo --> @@ -229,7 +171,7 @@ MathJax.Hub.Config({ - Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks + Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning
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    • -
  • 3D volumes of neurons
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  • More on GD and cost function
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  • An implementation of a Deep Neural Network
    • -
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  • Solving the ODE
    • -
  • Using neural network
    • -
  • Using a deep neural network
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  • Wrapping it up
    • @@ -396,7 +300,7 @@ for a FFNN to fulfill the universal approximation theorem
  • 26
  • 27
  • ...
    • -
  • 105
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs018.html b/doc/pub/NeuralNet/html/._NeuralNet-bs018.html index e0dbfb7f7..05f3d5517 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs018.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs018.html @@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source - + -Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks +Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning @@ -141,65 +141,7 @@ Automatically generated HTML file from DocOnce source ('Limitations of supervised learning with deep networks', 2, None, - '___sec65'), - ('Convolutional Neural Networks (recognizing images)', - 2, - None, - '___sec66'), - ('Regular NNs don’t scale well to full images', - 2, - None, - '___sec67'), - ('3D volumes of neurons', 2, None, '___sec68'), - ('Layers used to build CNNs', 2, None, '___sec69'), - ('Transforming images', 2, None, '___sec70'), - ('CNNs in brief', 2, None, '___sec71'), - ('CNNs in more detail, building convolutional neural networks in ' - 'Tensorflow and Keras', - 2, - None, - '___sec72'), - ('Setting it up', 2, None, '___sec73'), - ('The MNIST dataset again', 2, None, '___sec74'), - ('Strong correlations', 2, None, '___sec75'), - ('Layers of a CNN', 2, None, '___sec76'), - ('Systematic reduction', 2, None, '___sec77'), - ('Prerequisites: Collect and pre-process data', - 2, - None, - '___sec78'), - ('Importing Keras and Tensorflow', 2, None, '___sec79'), - ('Using TensorFlow backend', 2, None, '___sec80'), - ('Train the model', 2, None, '___sec81'), - ('Visualizing the results', 2, None, '___sec82'), - ('Running with Keras', 2, None, '___sec83'), - ('Final part', 2, None, '___sec84'), - ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86'), - ('Applications: solving ordinary differential equations with ' - 'Neural Networks', - 2, - None, - '___sec87'), - ('Trial solution', 2, None, '___sec88'), - ('More details', 2, None, '___sec89'), - ('Reformulating the problem', 2, None, '___sec90'), - ('Estimating errors', 2, None, '___sec91'), - ('Creating a simple Deep Neural Net', 2, None, '___sec92'), - ('Setting up the code, feed forward part', 2, None, '___sec93'), - ('Backpropagation', 2, None, '___sec94'), - ('Gradient Descent', 2, None, '___sec95'), - ('More on GD and cost function', 2, None, '___sec96'), - ('An implementation of a Deep Neural Network', - 2, - None, - '___sec97'), - ('The final parts of the code', 2, None, '___sec98'), - ('And adding Back propagation', 2, None, '___sec99'), - ('Solving the ODE', 2, None, '___sec100'), - ('Using neural network', 2, None, '___sec101'), - ('Using a deep neural network', 2, None, '___sec102'), - ('Wrapping it up', 2, None, '___sec103')]} + '___sec65')]} end of tocinfo --> @@ -229,7 +171,7 @@ MathJax.Hub.Config({ - Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks + Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning
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  • Regular NNs don’t scale well to full images
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  • 3D volumes of neurons
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  • Transforming images
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  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
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  • Strong correlations
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  • Visualizing the results
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  • Running with Keras
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  • Creating a simple Deep Neural Net
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  • Setting up the code, feed forward part
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  • Gradient Descent
    • -
  • More on GD and cost function
    • -
  • An implementation of a Deep Neural Network
    • -
  • The final parts of the code
    • -
  • And adding Back propagation
    • -
  • Solving the ODE
    • -
  • Using neural network
    • -
  • Using a deep neural network
    • -
  • Wrapping it up
    • @@ -404,7 +308,7 @@ $$
  • 27
  • 28
  • ...
    • -
  • 105
    • +
  • 67
  • »
    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs019.html b/doc/pub/NeuralNet/html/._NeuralNet-bs019.html index 9ec70eb01..7b0217935 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs019.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs019.html @@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source - + -Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks +Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning @@ -141,65 +141,7 @@ Automatically generated HTML file from DocOnce source ('Limitations of supervised learning with deep networks', 2, None, - '___sec65'), - ('Convolutional Neural Networks (recognizing images)', - 2, - None, - '___sec66'), - ('Regular NNs don’t scale well to full images', - 2, - None, - '___sec67'), - ('3D volumes of neurons', 2, None, '___sec68'), - ('Layers used to build CNNs', 2, None, '___sec69'), - ('Transforming images', 2, None, '___sec70'), - ('CNNs in brief', 2, None, '___sec71'), - ('CNNs in more detail, building convolutional neural networks in ' - 'Tensorflow and Keras', - 2, - None, - '___sec72'), - ('Setting it up', 2, None, '___sec73'), - ('The MNIST dataset again', 2, None, '___sec74'), - ('Strong correlations', 2, None, '___sec75'), - ('Layers of a CNN', 2, None, '___sec76'), - ('Systematic reduction', 2, None, '___sec77'), - ('Prerequisites: Collect and pre-process data', - 2, - None, - '___sec78'), - ('Importing Keras and Tensorflow', 2, None, '___sec79'), - ('Using TensorFlow backend', 2, None, '___sec80'), - ('Train the model', 2, None, '___sec81'), - ('Visualizing the results', 2, None, '___sec82'), - ('Running with Keras', 2, None, '___sec83'), - ('Final part', 2, None, '___sec84'), - ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86'), - ('Applications: solving ordinary differential equations with ' - 'Neural Networks', - 2, - None, - '___sec87'), - ('Trial solution', 2, None, '___sec88'), - ('More details', 2, None, '___sec89'), - ('Reformulating the problem', 2, None, '___sec90'), - ('Estimating errors', 2, None, '___sec91'), - ('Creating a simple Deep Neural Net', 2, None, '___sec92'), - ('Setting up the code, feed forward part', 2, None, '___sec93'), - ('Backpropagation', 2, None, '___sec94'), - ('Gradient Descent', 2, None, '___sec95'), - ('More on GD and cost function', 2, None, '___sec96'), - ('An implementation of a Deep Neural Network', - 2, - None, - '___sec97'), - ('The final parts of the code', 2, None, '___sec98'), - ('And adding Back propagation', 2, None, '___sec99'), - ('Solving the ODE', 2, None, '___sec100'), - ('Using neural network', 2, None, '___sec101'), - ('Using a deep neural network', 2, None, '___sec102'), - ('Wrapping it up', 2, None, '___sec103')]} + '___sec65')]} end of tocinfo --> @@ -229,7 +171,7 @@ MathJax.Hub.Config({ - Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks + Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning
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  • Regular NNs don’t scale well to full images
    • -
  • 3D volumes of neurons
    • -
  • Layers used to build CNNs
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  • Transforming images
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  • CNNs in brief
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  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
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  • Setting it up
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  • The MNIST dataset again
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  • Strong correlations
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  • Layers of a CNN
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  • Systematic reduction
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  • Visualizing the results
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  • Running with Keras
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  • Final part
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  • Final visualization
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  • Fun links
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  • Creating a simple Deep Neural Net
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  • Setting up the code, feed forward part
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    • -
  • Gradient Descent
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  • More on GD and cost function
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  • An implementation of a Deep Neural Network
    • -
  • The final parts of the code
    • -
  • And adding Back propagation
    • -
  • Solving the ODE
    • -
  • Using neural network
    • -
  • Using a deep neural network
    • -
  • Wrapping it up
    • @@ -466,7 +370,7 @@ plt.show()
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs020.html b/doc/pub/NeuralNet/html/._NeuralNet-bs020.html index 341255f5d..ac2f37123 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs020.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs020.html @@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source - + -Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks +Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning @@ -141,65 +141,7 @@ Automatically generated HTML file from DocOnce source ('Limitations of supervised learning with deep networks', 2, None, - '___sec65'), - ('Convolutional Neural Networks (recognizing images)', - 2, - None, - '___sec66'), - ('Regular NNs don’t scale well to full images', - 2, - None, - '___sec67'), - ('3D volumes of neurons', 2, None, '___sec68'), - ('Layers used to build CNNs', 2, None, '___sec69'), - ('Transforming images', 2, None, '___sec70'), - ('CNNs in brief', 2, None, '___sec71'), - ('CNNs in more detail, building convolutional neural networks in ' - 'Tensorflow and Keras', - 2, - None, - '___sec72'), - ('Setting it up', 2, None, '___sec73'), - ('The MNIST dataset again', 2, None, '___sec74'), - ('Strong correlations', 2, None, '___sec75'), - ('Layers of a CNN', 2, None, '___sec76'), - ('Systematic reduction', 2, None, '___sec77'), - ('Prerequisites: Collect and pre-process data', - 2, - None, - '___sec78'), - ('Importing Keras and Tensorflow', 2, None, '___sec79'), - ('Using TensorFlow backend', 2, None, '___sec80'), - ('Train the model', 2, None, '___sec81'), - ('Visualizing the results', 2, None, '___sec82'), - ('Running with Keras', 2, None, '___sec83'), - ('Final part', 2, None, '___sec84'), - ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86'), - ('Applications: solving ordinary differential equations with ' - 'Neural Networks', - 2, - None, - '___sec87'), - ('Trial solution', 2, None, '___sec88'), - ('More details', 2, None, '___sec89'), - ('Reformulating the problem', 2, None, '___sec90'), - ('Estimating errors', 2, None, '___sec91'), - ('Creating a simple Deep Neural Net', 2, None, '___sec92'), - ('Setting up the code, feed forward part', 2, None, '___sec93'), - ('Backpropagation', 2, None, '___sec94'), - ('Gradient Descent', 2, None, '___sec95'), - ('More on GD and cost function', 2, None, '___sec96'), - ('An implementation of a Deep Neural Network', - 2, - None, - '___sec97'), - ('The final parts of the code', 2, None, '___sec98'), - ('And adding Back propagation', 2, None, '___sec99'), - ('Solving the ODE', 2, None, '___sec100'), - ('Using neural network', 2, None, '___sec101'), - ('Using a deep neural network', 2, None, '___sec102'), - ('Wrapping it up', 2, None, '___sec103')]} + '___sec65')]} end of tocinfo --> @@ -229,7 +171,7 @@ MathJax.Hub.Config({ - Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks + Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning
    @@ -303,44 +245,6 @@ MathJax.Hub.Config({
  • Which activation function should we use?
  • A top-down perspective on Neural networks
  • Limitations of supervised learning with deep networks
    • -
  • Convolutional Neural Networks (recognizing images)
    • -
  • Regular NNs don’t scale well to full images
    • -
  • 3D volumes of neurons
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  • Transforming images
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  • The MNIST dataset again
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  • Strong correlations
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  • Importing Keras and Tensorflow
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  • Using TensorFlow backend
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  • Train the model
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  • Visualizing the results
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  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • Fun links
    • -
  • Applications: solving ordinary differential equations with Neural Networks
    • -
  • Trial solution
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  • More details
    • -
  • Reformulating the problem
    • -
  • Estimating errors
    • -
  • Creating a simple Deep Neural Net
    • -
  • Setting up the code, feed forward part
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  • Backpropagation
    • -
  • Gradient Descent
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  • More on GD and cost function
    • -
  • An implementation of a Deep Neural Network
    • -
  • The final parts of the code
    • -
  • And adding Back propagation
    • -
  • Solving the ODE
    • -
  • Using neural network
    • -
  • Using a deep neural network
    • -
  • Wrapping it up
    • @@ -417,7 +321,7 @@ like logistic regression or linear regression and their modifications on the oth
  • 29
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  • 67
  • »
    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs021.html b/doc/pub/NeuralNet/html/._NeuralNet-bs021.html index f608b5c1b..4ef6f7875 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs021.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs021.html @@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source - + -Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks +Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning @@ -141,65 +141,7 @@ Automatically generated HTML file from DocOnce source ('Limitations of supervised learning with deep networks', 2, None, - '___sec65'), - ('Convolutional Neural Networks (recognizing images)', - 2, - None, - '___sec66'), - ('Regular NNs don’t scale well to full images', - 2, - None, - '___sec67'), - ('3D volumes of neurons', 2, None, '___sec68'), - ('Layers used to build CNNs', 2, None, '___sec69'), - ('Transforming images', 2, None, '___sec70'), - ('CNNs in brief', 2, None, '___sec71'), - ('CNNs in more detail, building convolutional neural networks in ' - 'Tensorflow and Keras', - 2, - None, - '___sec72'), - ('Setting it up', 2, None, '___sec73'), - ('The MNIST dataset again', 2, None, '___sec74'), - ('Strong correlations', 2, None, '___sec75'), - ('Layers of a CNN', 2, None, '___sec76'), - ('Systematic reduction', 2, None, '___sec77'), - ('Prerequisites: Collect and pre-process data', - 2, - None, - '___sec78'), - ('Importing Keras and Tensorflow', 2, None, '___sec79'), - ('Using TensorFlow backend', 2, None, '___sec80'), - ('Train the model', 2, None, '___sec81'), - ('Visualizing the results', 2, None, '___sec82'), - ('Running with Keras', 2, None, '___sec83'), - ('Final part', 2, None, '___sec84'), - ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86'), - ('Applications: solving ordinary differential equations with ' - 'Neural Networks', - 2, - None, - '___sec87'), - ('Trial solution', 2, None, '___sec88'), - ('More details', 2, None, '___sec89'), - ('Reformulating the problem', 2, None, '___sec90'), - ('Estimating errors', 2, None, '___sec91'), - ('Creating a simple Deep Neural Net', 2, None, '___sec92'), - ('Setting up the code, feed forward part', 2, None, '___sec93'), - ('Backpropagation', 2, None, '___sec94'), - ('Gradient Descent', 2, None, '___sec95'), - ('More on GD and cost function', 2, None, '___sec96'), - ('An implementation of a Deep Neural Network', - 2, - None, - '___sec97'), - ('The final parts of the code', 2, None, '___sec98'), - ('And adding Back propagation', 2, None, '___sec99'), - ('Solving the ODE', 2, None, '___sec100'), - ('Using neural network', 2, None, '___sec101'), - ('Using a deep neural network', 2, None, '___sec102'), - ('Wrapping it up', 2, None, '___sec103')]} + '___sec65')]} end of tocinfo --> @@ -229,7 +171,7 @@ MathJax.Hub.Config({ - Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks + Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning
    @@ -303,44 +245,6 @@ MathJax.Hub.Config({
  • Which activation function should we use?
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  • Regular NNs don’t scale well to full images
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  • 3D volumes of neurons
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  • More on GD and cost function
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  • An implementation of a Deep Neural Network
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  • The final parts of the code
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  • And adding Back propagation
    • -
  • Solving the ODE
    • -
  • Using neural network
    • -
  • Using a deep neural network
    • -
  • Wrapping it up
    • @@ -407,7 +311,7 @@ the potential of being universal approximators.
  • 30
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  • Which activation function should we use?
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  • Regular NNs don’t scale well to full images
    • -
  • 3D volumes of neurons
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  • Transforming images
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  • Strong correlations
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  • Layers of a CNN
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  • Running with Keras
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  • Trial solution
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  • More on GD and cost function
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  • An implementation of a Deep Neural Network
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  • And adding Back propagation
    • -
  • Solving the ODE
    • -
  • Using neural network
    • -
  • Using a deep neural network
    • -
  • Wrapping it up
    • @@ -413,7 +317,7 @@ classes.
  • 31
  • 32
  • ...
    • -
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  • 3D volumes of neurons
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  • An implementation of a Deep Neural Network
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  • Solving the ODE
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  • Using neural network
    • -
  • Using a deep neural network
    • -
  • Wrapping it up
    • @@ -417,7 +321,7 @@ $$
  • 32
  • 33
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    • -
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs024.html b/doc/pub/NeuralNet/html/._NeuralNet-bs024.html index dcfbe41ea..31b961462 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs024.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs024.html @@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source - + -Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks +Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning @@ -141,65 +141,7 @@ Automatically generated HTML file from DocOnce source ('Limitations of supervised learning with deep networks', 2, None, - '___sec65'), - ('Convolutional Neural Networks (recognizing images)', - 2, - None, - '___sec66'), - ('Regular NNs don’t scale well to full images', - 2, - None, - '___sec67'), - ('3D volumes of neurons', 2, None, '___sec68'), - ('Layers used to build CNNs', 2, None, '___sec69'), - ('Transforming images', 2, None, '___sec70'), - ('CNNs in brief', 2, None, '___sec71'), - ('CNNs in more detail, building convolutional neural networks in ' - 'Tensorflow and Keras', - 2, - None, - '___sec72'), - ('Setting it up', 2, None, '___sec73'), - ('The MNIST dataset again', 2, None, '___sec74'), - ('Strong correlations', 2, None, '___sec75'), - ('Layers of a CNN', 2, None, '___sec76'), - ('Systematic reduction', 2, None, '___sec77'), - ('Prerequisites: Collect and pre-process data', - 2, - None, - '___sec78'), - ('Importing Keras and Tensorflow', 2, None, '___sec79'), - ('Using TensorFlow backend', 2, None, '___sec80'), - ('Train the model', 2, None, '___sec81'), - ('Visualizing the results', 2, None, '___sec82'), - ('Running with Keras', 2, None, '___sec83'), - ('Final part', 2, None, '___sec84'), - ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86'), - ('Applications: solving ordinary differential equations with ' - 'Neural Networks', - 2, - None, - '___sec87'), - ('Trial solution', 2, None, '___sec88'), - ('More details', 2, None, '___sec89'), - ('Reformulating the problem', 2, None, '___sec90'), - ('Estimating errors', 2, None, '___sec91'), - ('Creating a simple Deep Neural Net', 2, None, '___sec92'), - ('Setting up the code, feed forward part', 2, None, '___sec93'), - ('Backpropagation', 2, None, '___sec94'), - ('Gradient Descent', 2, None, '___sec95'), - ('More on GD and cost function', 2, None, '___sec96'), - ('An implementation of a Deep Neural Network', - 2, - None, - '___sec97'), - ('The final parts of the code', 2, None, '___sec98'), - ('And adding Back propagation', 2, None, '___sec99'), - ('Solving the ODE', 2, None, '___sec100'), - ('Using neural network', 2, None, '___sec101'), - ('Using a deep neural network', 2, None, '___sec102'), - ('Wrapping it up', 2, None, '___sec103')]} + '___sec65')]} end of tocinfo --> @@ -229,7 +171,7 @@ MathJax.Hub.Config({ - Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks + Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning
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  • Regular NNs don’t scale well to full images
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  • 3D volumes of neurons
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  • More on GD and cost function
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  • An implementation of a Deep Neural Network
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  • The final parts of the code
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    • -
  • Solving the ODE
    • -
  • Using neural network
    • -
  • Using a deep neural network
    • -
  • Wrapping it up
    • @@ -401,7 +305,7 @@ $$
  • 33
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  • ...
    • -
  • 105
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    @@ -303,44 +245,6 @@ MathJax.Hub.Config({
  • Which activation function should we use?
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  • Regular NNs don’t scale well to full images
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  • 3D volumes of neurons
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  • The MNIST dataset again
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  • Strong correlations
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  • Layers of a CNN
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  • Importing Keras and Tensorflow
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  • Visualizing the results
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  • Running with Keras
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  • Final part
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  • Final visualization
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  • Fun links
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  • Trial solution
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  • More details
    • -
  • Reformulating the problem
    • -
  • Estimating errors
    • -
  • Creating a simple Deep Neural Net
    • -
  • Setting up the code, feed forward part
    • -
  • Backpropagation
    • -
  • Gradient Descent
    • -
  • More on GD and cost function
    • -
  • An implementation of a Deep Neural Network
    • -
  • The final parts of the code
    • -
  • And adding Back propagation
    • -
  • Solving the ODE
    • -
  • Using neural network
    • -
  • Using a deep neural network
    • -
  • Wrapping it up
    • @@ -404,7 +308,7 @@ $$
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs026.html b/doc/pub/NeuralNet/html/._NeuralNet-bs026.html index 085bdb252..6a8c35a12 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs026.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs026.html @@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source - + -Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks +Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning @@ -141,65 +141,7 @@ Automatically generated HTML file from DocOnce source ('Limitations of supervised learning with deep networks', 2, None, - '___sec65'), - ('Convolutional Neural Networks (recognizing images)', - 2, - None, - '___sec66'), - ('Regular NNs don’t scale well to full images', - 2, - None, - '___sec67'), - ('3D volumes of neurons', 2, None, '___sec68'), - ('Layers used to build CNNs', 2, None, '___sec69'), - ('Transforming images', 2, None, '___sec70'), - ('CNNs in brief', 2, None, '___sec71'), - ('CNNs in more detail, building convolutional neural networks in ' - 'Tensorflow and Keras', - 2, - None, - '___sec72'), - ('Setting it up', 2, None, '___sec73'), - ('The MNIST dataset again', 2, None, '___sec74'), - ('Strong correlations', 2, None, '___sec75'), - ('Layers of a CNN', 2, None, '___sec76'), - ('Systematic reduction', 2, None, '___sec77'), - ('Prerequisites: Collect and pre-process data', - 2, - None, - '___sec78'), - ('Importing Keras and Tensorflow', 2, None, '___sec79'), - ('Using TensorFlow backend', 2, None, '___sec80'), - ('Train the model', 2, None, '___sec81'), - ('Visualizing the results', 2, None, '___sec82'), - ('Running with Keras', 2, None, '___sec83'), - ('Final part', 2, None, '___sec84'), - ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86'), - ('Applications: solving ordinary differential equations with ' - 'Neural Networks', - 2, - None, - '___sec87'), - ('Trial solution', 2, None, '___sec88'), - ('More details', 2, None, '___sec89'), - ('Reformulating the problem', 2, None, '___sec90'), - ('Estimating errors', 2, None, '___sec91'), - ('Creating a simple Deep Neural Net', 2, None, '___sec92'), - ('Setting up the code, feed forward part', 2, None, '___sec93'), - ('Backpropagation', 2, None, '___sec94'), - ('Gradient Descent', 2, None, '___sec95'), - ('More on GD and cost function', 2, None, '___sec96'), - ('An implementation of a Deep Neural Network', - 2, - None, - '___sec97'), - ('The final parts of the code', 2, None, '___sec98'), - ('And adding Back propagation', 2, None, '___sec99'), - ('Solving the ODE', 2, None, '___sec100'), - ('Using neural network', 2, None, '___sec101'), - ('Using a deep neural network', 2, None, '___sec102'), - ('Wrapping it up', 2, None, '___sec103')]} + '___sec65')]} end of tocinfo --> @@ -229,7 +171,7 @@ MathJax.Hub.Config({ - Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks + Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning
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  • Regular NNs don’t scale well to full images
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  • 3D volumes of neurons
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  • Transforming images
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  • Running with Keras
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  • Trial solution
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  • Estimating errors
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  • Creating a simple Deep Neural Net
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  • Setting up the code, feed forward part
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  • Backpropagation
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  • Gradient Descent
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  • More on GD and cost function
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  • An implementation of a Deep Neural Network
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  • The final parts of the code
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  • And adding Back propagation
    • -
  • Solving the ODE
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  • Using neural network
    • -
  • Using a deep neural network
    • -
  • Wrapping it up
    • @@ -429,7 +333,7 @@ $$
  • 35
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs027.html b/doc/pub/NeuralNet/html/._NeuralNet-bs027.html index 960bc40d5..24e9f70de 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs027.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs027.html @@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source - + -Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks +Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning @@ -141,65 +141,7 @@ Automatically generated HTML file from DocOnce source ('Limitations of supervised learning with deep networks', 2, None, - '___sec65'), - ('Convolutional Neural Networks (recognizing images)', - 2, - None, - '___sec66'), - ('Regular NNs don’t scale well to full images', - 2, - None, - '___sec67'), - ('3D volumes of neurons', 2, None, '___sec68'), - ('Layers used to build CNNs', 2, None, '___sec69'), - ('Transforming images', 2, None, '___sec70'), - ('CNNs in brief', 2, None, '___sec71'), - ('CNNs in more detail, building convolutional neural networks in ' - 'Tensorflow and Keras', - 2, - None, - '___sec72'), - ('Setting it up', 2, None, '___sec73'), - ('The MNIST dataset again', 2, None, '___sec74'), - ('Strong correlations', 2, None, '___sec75'), - ('Layers of a CNN', 2, None, '___sec76'), - ('Systematic reduction', 2, None, '___sec77'), - ('Prerequisites: Collect and pre-process data', - 2, - None, - '___sec78'), - ('Importing Keras and Tensorflow', 2, None, '___sec79'), - ('Using TensorFlow backend', 2, None, '___sec80'), - ('Train the model', 2, None, '___sec81'), - ('Visualizing the results', 2, None, '___sec82'), - ('Running with Keras', 2, None, '___sec83'), - ('Final part', 2, None, '___sec84'), - ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86'), - ('Applications: solving ordinary differential equations with ' - 'Neural Networks', - 2, - None, - '___sec87'), - ('Trial solution', 2, None, '___sec88'), - ('More details', 2, None, '___sec89'), - ('Reformulating the problem', 2, None, '___sec90'), - ('Estimating errors', 2, None, '___sec91'), - ('Creating a simple Deep Neural Net', 2, None, '___sec92'), - ('Setting up the code, feed forward part', 2, None, '___sec93'), - ('Backpropagation', 2, None, '___sec94'), - ('Gradient Descent', 2, None, '___sec95'), - ('More on GD and cost function', 2, None, '___sec96'), - ('An implementation of a Deep Neural Network', - 2, - None, - '___sec97'), - ('The final parts of the code', 2, None, '___sec98'), - ('And adding Back propagation', 2, None, '___sec99'), - ('Solving the ODE', 2, None, '___sec100'), - ('Using neural network', 2, None, '___sec101'), - ('Using a deep neural network', 2, None, '___sec102'), - ('Wrapping it up', 2, None, '___sec103')]} + '___sec65')]} end of tocinfo --> @@ -229,7 +171,7 @@ MathJax.Hub.Config({ - Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks + Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning
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  • Which activation function should we use?
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  • Regular NNs don’t scale well to full images
    • -
  • 3D volumes of neurons
    • -
  • Layers used to build CNNs
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  • Transforming images
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  • CNNs in brief
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
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  • Setting it up
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  • Strong correlations
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  • Layers of a CNN
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  • Running with Keras
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  • Trial solution
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  • Gradient Descent
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  • More on GD and cost function
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  • An implementation of a Deep Neural Network
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  • The final parts of the code
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  • And adding Back propagation
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  • Solving the ODE
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  • Using neural network
    • -
  • Using a deep neural network
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  • Wrapping it up
    • @@ -396,7 +300,7 @@ That is, the error \( \delta_j^L \) is exactly equal to the rate of change of th
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs028.html b/doc/pub/NeuralNet/html/._NeuralNet-bs028.html index 615803cac..96c0fd207 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs028.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs028.html @@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source - + -Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks +Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning @@ -141,65 +141,7 @@ Automatically generated HTML file from DocOnce source ('Limitations of supervised learning with deep networks', 2, None, - '___sec65'), - ('Convolutional Neural Networks (recognizing images)', - 2, - None, - '___sec66'), - ('Regular NNs don’t scale well to full images', - 2, - None, - '___sec67'), - ('3D volumes of neurons', 2, None, '___sec68'), - ('Layers used to build CNNs', 2, None, '___sec69'), - ('Transforming images', 2, None, '___sec70'), - ('CNNs in brief', 2, None, '___sec71'), - ('CNNs in more detail, building convolutional neural networks in ' - 'Tensorflow and Keras', - 2, - None, - '___sec72'), - ('Setting it up', 2, None, '___sec73'), - ('The MNIST dataset again', 2, None, '___sec74'), - ('Strong correlations', 2, None, '___sec75'), - ('Layers of a CNN', 2, None, '___sec76'), - ('Systematic reduction', 2, None, '___sec77'), - ('Prerequisites: Collect and pre-process data', - 2, - None, - '___sec78'), - ('Importing Keras and Tensorflow', 2, None, '___sec79'), - ('Using TensorFlow backend', 2, None, '___sec80'), - ('Train the model', 2, None, '___sec81'), - ('Visualizing the results', 2, None, '___sec82'), - ('Running with Keras', 2, None, '___sec83'), - ('Final part', 2, None, '___sec84'), - ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86'), - ('Applications: solving ordinary differential equations with ' - 'Neural Networks', - 2, - None, - '___sec87'), - ('Trial solution', 2, None, '___sec88'), - ('More details', 2, None, '___sec89'), - ('Reformulating the problem', 2, None, '___sec90'), - ('Estimating errors', 2, None, '___sec91'), - ('Creating a simple Deep Neural Net', 2, None, '___sec92'), - ('Setting up the code, feed forward part', 2, None, '___sec93'), - ('Backpropagation', 2, None, '___sec94'), - ('Gradient Descent', 2, None, '___sec95'), - ('More on GD and cost function', 2, None, '___sec96'), - ('An implementation of a Deep Neural Network', - 2, - None, - '___sec97'), - ('The final parts of the code', 2, None, '___sec98'), - ('And adding Back propagation', 2, None, '___sec99'), - ('Solving the ODE', 2, None, '___sec100'), - ('Using neural network', 2, None, '___sec101'), - ('Using a deep neural network', 2, None, '___sec102'), - ('Wrapping it up', 2, None, '___sec103')]} + '___sec65')]} end of tocinfo --> @@ -229,7 +171,7 @@ MathJax.Hub.Config({ - Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks + Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning
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  • Which activation function should we use?
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  • Regular NNs don’t scale well to full images
    • -
  • 3D volumes of neurons
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  • Transforming images
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  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
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  • Systematic reduction
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  • Using TensorFlow backend
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  • Visualizing the results
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  • Running with Keras
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  • Gradient Descent
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  • More on GD and cost function
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  • An implementation of a Deep Neural Network
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  • The final parts of the code
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  • And adding Back propagation
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  • Solving the ODE
    • -
  • Using neural network
    • -
  • Using a deep neural network
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  • Wrapping it up
    • @@ -441,7 +345,7 @@ one \( L-1 \) in terms of the errors in the final output layer.
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs029.html b/doc/pub/NeuralNet/html/._NeuralNet-bs029.html index d4d743b53..4560da562 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs029.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs029.html @@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source - + -Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks +Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning @@ -141,65 +141,7 @@ Automatically generated HTML file from DocOnce source ('Limitations of supervised learning with deep networks', 2, None, - '___sec65'), - ('Convolutional Neural Networks (recognizing images)', - 2, - None, - '___sec66'), - ('Regular NNs don’t scale well to full images', - 2, - None, - '___sec67'), - ('3D volumes of neurons', 2, None, '___sec68'), - ('Layers used to build CNNs', 2, None, '___sec69'), - ('Transforming images', 2, None, '___sec70'), - ('CNNs in brief', 2, None, '___sec71'), - ('CNNs in more detail, building convolutional neural networks in ' - 'Tensorflow and Keras', - 2, - None, - '___sec72'), - ('Setting it up', 2, None, '___sec73'), - ('The MNIST dataset again', 2, None, '___sec74'), - ('Strong correlations', 2, None, '___sec75'), - ('Layers of a CNN', 2, None, '___sec76'), - ('Systematic reduction', 2, None, '___sec77'), - ('Prerequisites: Collect and pre-process data', - 2, - None, - '___sec78'), - ('Importing Keras and Tensorflow', 2, None, '___sec79'), - ('Using TensorFlow backend', 2, None, '___sec80'), - ('Train the model', 2, None, '___sec81'), - ('Visualizing the results', 2, None, '___sec82'), - ('Running with Keras', 2, None, '___sec83'), - ('Final part', 2, None, '___sec84'), - ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86'), - ('Applications: solving ordinary differential equations with ' - 'Neural Networks', - 2, - None, - '___sec87'), - ('Trial solution', 2, None, '___sec88'), - ('More details', 2, None, '___sec89'), - ('Reformulating the problem', 2, None, '___sec90'), - ('Estimating errors', 2, None, '___sec91'), - ('Creating a simple Deep Neural Net', 2, None, '___sec92'), - ('Setting up the code, feed forward part', 2, None, '___sec93'), - ('Backpropagation', 2, None, '___sec94'), - ('Gradient Descent', 2, None, '___sec95'), - ('More on GD and cost function', 2, None, '___sec96'), - ('An implementation of a Deep Neural Network', - 2, - None, - '___sec97'), - ('The final parts of the code', 2, None, '___sec98'), - ('And adding Back propagation', 2, None, '___sec99'), - ('Solving the ODE', 2, None, '___sec100'), - ('Using neural network', 2, None, '___sec101'), - ('Using a deep neural network', 2, None, '___sec102'), - ('Wrapping it up', 2, None, '___sec103')]} + '___sec65')]} end of tocinfo --> @@ -229,7 +171,7 @@ MathJax.Hub.Config({ - Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks + Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning
    @@ -303,44 +245,6 @@ MathJax.Hub.Config({
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    • -
  • Regular NNs don’t scale well to full images
    • -
  • 3D volumes of neurons
    • -
  • Layers used to build CNNs
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  • Transforming images
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  • CNNs in brief
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  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
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  • Setting it up
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  • The MNIST dataset again
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  • Strong correlations
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  • Layers of a CNN
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  • Systematic reduction
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  • Running with Keras
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  • Gradient Descent
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  • More on GD and cost function
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  • An implementation of a Deep Neural Network
    • -
  • The final parts of the code
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  • And adding Back propagation
    • -
  • Solving the ODE
    • -
  • Using neural network
    • -
  • Using a deep neural network
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  • Wrapping it up
    • @@ -411,7 +315,7 @@ We are now ready to set up the algorithm for back propagation and learning the w
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs030.html b/doc/pub/NeuralNet/html/._NeuralNet-bs030.html index 3b3091423..9fa403205 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs030.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs030.html @@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source - + -Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks +Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning @@ -141,65 +141,7 @@ Automatically generated HTML file from DocOnce source ('Limitations of supervised learning with deep networks', 2, None, - '___sec65'), - ('Convolutional Neural Networks (recognizing images)', - 2, - None, - '___sec66'), - ('Regular NNs don’t scale well to full images', - 2, - None, - '___sec67'), - ('3D volumes of neurons', 2, None, '___sec68'), - ('Layers used to build CNNs', 2, None, '___sec69'), - ('Transforming images', 2, None, '___sec70'), - ('CNNs in brief', 2, None, '___sec71'), - ('CNNs in more detail, building convolutional neural networks in ' - 'Tensorflow and Keras', - 2, - None, - '___sec72'), - ('Setting it up', 2, None, '___sec73'), - ('The MNIST dataset again', 2, None, '___sec74'), - ('Strong correlations', 2, None, '___sec75'), - ('Layers of a CNN', 2, None, '___sec76'), - ('Systematic reduction', 2, None, '___sec77'), - ('Prerequisites: Collect and pre-process data', - 2, - None, - '___sec78'), - ('Importing Keras and Tensorflow', 2, None, '___sec79'), - ('Using TensorFlow backend', 2, None, '___sec80'), - ('Train the model', 2, None, '___sec81'), - ('Visualizing the results', 2, None, '___sec82'), - ('Running with Keras', 2, None, '___sec83'), - ('Final part', 2, None, '___sec84'), - ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86'), - ('Applications: solving ordinary differential equations with ' - 'Neural Networks', - 2, - None, - '___sec87'), - ('Trial solution', 2, None, '___sec88'), - ('More details', 2, None, '___sec89'), - ('Reformulating the problem', 2, None, '___sec90'), - ('Estimating errors', 2, None, '___sec91'), - ('Creating a simple Deep Neural Net', 2, None, '___sec92'), - ('Setting up the code, feed forward part', 2, None, '___sec93'), - ('Backpropagation', 2, None, '___sec94'), - ('Gradient Descent', 2, None, '___sec95'), - ('More on GD and cost function', 2, None, '___sec96'), - ('An implementation of a Deep Neural Network', - 2, - None, - '___sec97'), - ('The final parts of the code', 2, None, '___sec98'), - ('And adding Back propagation', 2, None, '___sec99'), - ('Solving the ODE', 2, None, '___sec100'), - ('Using neural network', 2, None, '___sec101'), - ('Using a deep neural network', 2, None, '___sec102'), - ('Wrapping it up', 2, None, '___sec103')]} + '___sec65')]} end of tocinfo --> @@ -229,7 +171,7 @@ MathJax.Hub.Config({ - Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks + Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning
    @@ -303,44 +245,6 @@ MathJax.Hub.Config({
  • Which activation function should we use?
  • A top-down perspective on Neural networks
  • Limitations of supervised learning with deep networks
    • -
  • Convolutional Neural Networks (recognizing images)
    • -
  • Regular NNs don’t scale well to full images
    • -
  • 3D volumes of neurons
    • -
  • Layers used to build CNNs
    • -
  • Transforming images
    • -
  • CNNs in brief
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Using TensorFlow backend
    • -
  • Train the model
    • -
  • Visualizing the results
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • Fun links
    • -
  • Applications: solving ordinary differential equations with Neural Networks
    • -
  • Trial solution
    • -
  • More details
    • -
  • Reformulating the problem
    • -
  • Estimating errors
    • -
  • Creating a simple Deep Neural Net
    • -
  • Setting up the code, feed forward part
    • -
  • Backpropagation
    • -
  • Gradient Descent
    • -
  • More on GD and cost function
    • -
  • An implementation of a Deep Neural Network
    • -
  • The final parts of the code
    • -
  • And adding Back propagation
    • -
  • Solving the ODE
    • -
  • Using neural network
    • -
  • Using a deep neural network
    • -
  • Wrapping it up
    • @@ -455,7 +359,7 @@ Here it is convenient to use stochastic gradient descent (see the examples below
  • 39
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs031.html b/doc/pub/NeuralNet/html/._NeuralNet-bs031.html index 12100ebaf..f2a123a34 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs031.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs031.html @@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source - + -Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks +Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning @@ -141,65 +141,7 @@ Automatically generated HTML file from DocOnce source ('Limitations of supervised learning with deep networks', 2, None, - '___sec65'), - ('Convolutional Neural Networks (recognizing images)', - 2, - None, - '___sec66'), - ('Regular NNs don’t scale well to full images', - 2, - None, - '___sec67'), - ('3D volumes of neurons', 2, None, '___sec68'), - ('Layers used to build CNNs', 2, None, '___sec69'), - ('Transforming images', 2, None, '___sec70'), - ('CNNs in brief', 2, None, '___sec71'), - ('CNNs in more detail, building convolutional neural networks in ' - 'Tensorflow and Keras', - 2, - None, - '___sec72'), - ('Setting it up', 2, None, '___sec73'), - ('The MNIST dataset again', 2, None, '___sec74'), - ('Strong correlations', 2, None, '___sec75'), - ('Layers of a CNN', 2, None, '___sec76'), - ('Systematic reduction', 2, None, '___sec77'), - ('Prerequisites: Collect and pre-process data', - 2, - None, - '___sec78'), - ('Importing Keras and Tensorflow', 2, None, '___sec79'), - ('Using TensorFlow backend', 2, None, '___sec80'), - ('Train the model', 2, None, '___sec81'), - ('Visualizing the results', 2, None, '___sec82'), - ('Running with Keras', 2, None, '___sec83'), - ('Final part', 2, None, '___sec84'), - ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86'), - ('Applications: solving ordinary differential equations with ' - 'Neural Networks', - 2, - None, - '___sec87'), - ('Trial solution', 2, None, '___sec88'), - ('More details', 2, None, '___sec89'), - ('Reformulating the problem', 2, None, '___sec90'), - ('Estimating errors', 2, None, '___sec91'), - ('Creating a simple Deep Neural Net', 2, None, '___sec92'), - ('Setting up the code, feed forward part', 2, None, '___sec93'), - ('Backpropagation', 2, None, '___sec94'), - ('Gradient Descent', 2, None, '___sec95'), - ('More on GD and cost function', 2, None, '___sec96'), - ('An implementation of a Deep Neural Network', - 2, - None, - '___sec97'), - ('The final parts of the code', 2, None, '___sec98'), - ('And adding Back propagation', 2, None, '___sec99'), - ('Solving the ODE', 2, None, '___sec100'), - ('Using neural network', 2, None, '___sec101'), - ('Using a deep neural network', 2, None, '___sec102'), - ('Wrapping it up', 2, None, '___sec103')]} + '___sec65')]} end of tocinfo --> @@ -229,7 +171,7 @@ MathJax.Hub.Config({ - Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks + Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning
    @@ -303,44 +245,6 @@ MathJax.Hub.Config({
  • Which activation function should we use?
  • A top-down perspective on Neural networks
  • Limitations of supervised learning with deep networks
    • -
  • Convolutional Neural Networks (recognizing images)
    • -
  • Regular NNs don’t scale well to full images
    • -
  • 3D volumes of neurons
    • -
  • Layers used to build CNNs
    • -
  • Transforming images
    • -
  • CNNs in brief
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
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  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
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  • Systematic reduction
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  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Using TensorFlow backend
    • -
  • Train the model
    • -
  • Visualizing the results
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • Fun links
    • -
  • Applications: solving ordinary differential equations with Neural Networks
    • -
  • Trial solution
    • -
  • More details
    • -
  • Reformulating the problem
    • -
  • Estimating errors
    • -
  • Creating a simple Deep Neural Net
    • -
  • Setting up the code, feed forward part
    • -
  • Backpropagation
    • -
  • Gradient Descent
    • -
  • More on GD and cost function
    • -
  • An implementation of a Deep Neural Network
    • -
  • The final parts of the code
    • -
  • And adding Back propagation
    • -
  • Solving the ODE
    • -
  • Using neural network
    • -
  • Using a deep neural network
    • -
  • Wrapping it up
    • @@ -417,7 +321,7 @@ of our network.
  • 40
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  • ...
    • -
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    • +
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs032.html b/doc/pub/NeuralNet/html/._NeuralNet-bs032.html index 18ff4402b..c342b3a96 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs032.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs032.html @@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source - + -Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks +Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning @@ -141,65 +141,7 @@ Automatically generated HTML file from DocOnce source ('Limitations of supervised learning with deep networks', 2, None, - '___sec65'), - ('Convolutional Neural Networks (recognizing images)', - 2, - None, - '___sec66'), - ('Regular NNs don’t scale well to full images', - 2, - None, - '___sec67'), - ('3D volumes of neurons', 2, None, '___sec68'), - ('Layers used to build CNNs', 2, None, '___sec69'), - ('Transforming images', 2, None, '___sec70'), - ('CNNs in brief', 2, None, '___sec71'), - ('CNNs in more detail, building convolutional neural networks in ' - 'Tensorflow and Keras', - 2, - None, - '___sec72'), - ('Setting it up', 2, None, '___sec73'), - ('The MNIST dataset again', 2, None, '___sec74'), - ('Strong correlations', 2, None, '___sec75'), - ('Layers of a CNN', 2, None, '___sec76'), - ('Systematic reduction', 2, None, '___sec77'), - ('Prerequisites: Collect and pre-process data', - 2, - None, - '___sec78'), - ('Importing Keras and Tensorflow', 2, None, '___sec79'), - ('Using TensorFlow backend', 2, None, '___sec80'), - ('Train the model', 2, None, '___sec81'), - ('Visualizing the results', 2, None, '___sec82'), - ('Running with Keras', 2, None, '___sec83'), - ('Final part', 2, None, '___sec84'), - ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86'), - ('Applications: solving ordinary differential equations with ' - 'Neural Networks', - 2, - None, - '___sec87'), - ('Trial solution', 2, None, '___sec88'), - ('More details', 2, None, '___sec89'), - ('Reformulating the problem', 2, None, '___sec90'), - ('Estimating errors', 2, None, '___sec91'), - ('Creating a simple Deep Neural Net', 2, None, '___sec92'), - ('Setting up the code, feed forward part', 2, None, '___sec93'), - ('Backpropagation', 2, None, '___sec94'), - ('Gradient Descent', 2, None, '___sec95'), - ('More on GD and cost function', 2, None, '___sec96'), - ('An implementation of a Deep Neural Network', - 2, - None, - '___sec97'), - ('The final parts of the code', 2, None, '___sec98'), - ('And adding Back propagation', 2, None, '___sec99'), - ('Solving the ODE', 2, None, '___sec100'), - ('Using neural network', 2, None, '___sec101'), - ('Using a deep neural network', 2, None, '___sec102'), - ('Wrapping it up', 2, None, '___sec103')]} + '___sec65')]} end of tocinfo --> @@ -229,7 +171,7 @@ MathJax.Hub.Config({ - Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks + Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning
    @@ -303,44 +245,6 @@ MathJax.Hub.Config({
  • Which activation function should we use?
  • A top-down perspective on Neural networks
  • Limitations of supervised learning with deep networks
    • -
  • Convolutional Neural Networks (recognizing images)
    • -
  • Regular NNs don’t scale well to full images
    • -
  • 3D volumes of neurons
    • -
  • Layers used to build CNNs
    • -
  • Transforming images
    • -
  • CNNs in brief
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Using TensorFlow backend
    • -
  • Train the model
    • -
  • Visualizing the results
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • Fun links
    • -
  • Applications: solving ordinary differential equations with Neural Networks
    • -
  • Trial solution
    • -
  • More details
    • -
  • Reformulating the problem
    • -
  • Estimating errors
    • -
  • Creating a simple Deep Neural Net
    • -
  • Setting up the code, feed forward part
    • -
  • Backpropagation
    • -
  • Gradient Descent
    • -
  • More on GD and cost function
    • -
  • An implementation of a Deep Neural Network
    • -
  • The final parts of the code
    • -
  • And adding Back propagation
    • -
  • Solving the ODE
    • -
  • Using neural network
    • -
  • Using a deep neural network
    • -
  • Wrapping it up
    • @@ -439,7 +343,7 @@ The back propagation equations need now only a small change, namely the definiti
  • 41
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs033.html b/doc/pub/NeuralNet/html/._NeuralNet-bs033.html index befe0650c..ba5a0b790 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs033.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs033.html @@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source - + -Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks +Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning @@ -141,65 +141,7 @@ Automatically generated HTML file from DocOnce source ('Limitations of supervised learning with deep networks', 2, None, - '___sec65'), - ('Convolutional Neural Networks (recognizing images)', - 2, - None, - '___sec66'), - ('Regular NNs don’t scale well to full images', - 2, - None, - '___sec67'), - ('3D volumes of neurons', 2, None, '___sec68'), - ('Layers used to build CNNs', 2, None, '___sec69'), - ('Transforming images', 2, None, '___sec70'), - ('CNNs in brief', 2, None, '___sec71'), - ('CNNs in more detail, building convolutional neural networks in ' - 'Tensorflow and Keras', - 2, - None, - '___sec72'), - ('Setting it up', 2, None, '___sec73'), - ('The MNIST dataset again', 2, None, '___sec74'), - ('Strong correlations', 2, None, '___sec75'), - ('Layers of a CNN', 2, None, '___sec76'), - ('Systematic reduction', 2, None, '___sec77'), - ('Prerequisites: Collect and pre-process data', - 2, - None, - '___sec78'), - ('Importing Keras and Tensorflow', 2, None, '___sec79'), - ('Using TensorFlow backend', 2, None, '___sec80'), - ('Train the model', 2, None, '___sec81'), - ('Visualizing the results', 2, None, '___sec82'), - ('Running with Keras', 2, None, '___sec83'), - ('Final part', 2, None, '___sec84'), - ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86'), - ('Applications: solving ordinary differential equations with ' - 'Neural Networks', - 2, - None, - '___sec87'), - ('Trial solution', 2, None, '___sec88'), - ('More details', 2, None, '___sec89'), - ('Reformulating the problem', 2, None, '___sec90'), - ('Estimating errors', 2, None, '___sec91'), - ('Creating a simple Deep Neural Net', 2, None, '___sec92'), - ('Setting up the code, feed forward part', 2, None, '___sec93'), - ('Backpropagation', 2, None, '___sec94'), - ('Gradient Descent', 2, None, '___sec95'), - ('More on GD and cost function', 2, None, '___sec96'), - ('An implementation of a Deep Neural Network', - 2, - None, - '___sec97'), - ('The final parts of the code', 2, None, '___sec98'), - ('And adding Back propagation', 2, None, '___sec99'), - ('Solving the ODE', 2, None, '___sec100'), - ('Using neural network', 2, None, '___sec101'), - ('Using a deep neural network', 2, None, '___sec102'), - ('Wrapping it up', 2, None, '___sec103')]} + '___sec65')]} end of tocinfo --> @@ -229,7 +171,7 @@ MathJax.Hub.Config({ - Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks + Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning
    @@ -303,44 +245,6 @@ MathJax.Hub.Config({
  • Which activation function should we use?
  • A top-down perspective on Neural networks
  • Limitations of supervised learning with deep networks
    • -
  • Convolutional Neural Networks (recognizing images)
    • -
  • Regular NNs don’t scale well to full images
    • -
  • 3D volumes of neurons
    • -
  • Layers used to build CNNs
    • -
  • Transforming images
    • -
  • CNNs in brief
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Using TensorFlow backend
    • -
  • Train the model
    • -
  • Visualizing the results
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • Fun links
    • -
  • Applications: solving ordinary differential equations with Neural Networks
    • -
  • Trial solution
    • -
  • More details
    • -
  • Reformulating the problem
    • -
  • Estimating errors
    • -
  • Creating a simple Deep Neural Net
    • -
  • Setting up the code, feed forward part
    • -
  • Backpropagation
    • -
  • Gradient Descent
    • -
  • More on GD and cost function
    • -
  • An implementation of a Deep Neural Network
    • -
  • The final parts of the code
    • -
  • And adding Back propagation
    • -
  • Solving the ODE
    • -
  • Using neural network
    • -
  • Using a deep neural network
    • -
  • Wrapping it up
    • @@ -427,7 +331,7 @@ In case we use another activation function than the logistic one, we need to eva
  • 42
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs034.html b/doc/pub/NeuralNet/html/._NeuralNet-bs034.html index f55858ae2..ae3b857e8 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs034.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs034.html @@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source - + -Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks +Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning @@ -141,65 +141,7 @@ Automatically generated HTML file from DocOnce source ('Limitations of supervised learning with deep networks', 2, None, - '___sec65'), - ('Convolutional Neural Networks (recognizing images)', - 2, - None, - '___sec66'), - ('Regular NNs don’t scale well to full images', - 2, - None, - '___sec67'), - ('3D volumes of neurons', 2, None, '___sec68'), - ('Layers used to build CNNs', 2, None, '___sec69'), - ('Transforming images', 2, None, '___sec70'), - ('CNNs in brief', 2, None, '___sec71'), - ('CNNs in more detail, building convolutional neural networks in ' - 'Tensorflow and Keras', - 2, - None, - '___sec72'), - ('Setting it up', 2, None, '___sec73'), - ('The MNIST dataset again', 2, None, '___sec74'), - ('Strong correlations', 2, None, '___sec75'), - ('Layers of a CNN', 2, None, '___sec76'), - ('Systematic reduction', 2, None, '___sec77'), - ('Prerequisites: Collect and pre-process data', - 2, - None, - '___sec78'), - ('Importing Keras and Tensorflow', 2, None, '___sec79'), - ('Using TensorFlow backend', 2, None, '___sec80'), - ('Train the model', 2, None, '___sec81'), - ('Visualizing the results', 2, None, '___sec82'), - ('Running with Keras', 2, None, '___sec83'), - ('Final part', 2, None, '___sec84'), - ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86'), - ('Applications: solving ordinary differential equations with ' - 'Neural Networks', - 2, - None, - '___sec87'), - ('Trial solution', 2, None, '___sec88'), - ('More details', 2, None, '___sec89'), - ('Reformulating the problem', 2, None, '___sec90'), - ('Estimating errors', 2, None, '___sec91'), - ('Creating a simple Deep Neural Net', 2, None, '___sec92'), - ('Setting up the code, feed forward part', 2, None, '___sec93'), - ('Backpropagation', 2, None, '___sec94'), - ('Gradient Descent', 2, None, '___sec95'), - ('More on GD and cost function', 2, None, '___sec96'), - ('An implementation of a Deep Neural Network', - 2, - None, - '___sec97'), - ('The final parts of the code', 2, None, '___sec98'), - ('And adding Back propagation', 2, None, '___sec99'), - ('Solving the ODE', 2, None, '___sec100'), - ('Using neural network', 2, None, '___sec101'), - ('Using a deep neural network', 2, None, '___sec102'), - ('Wrapping it up', 2, None, '___sec103')]} + '___sec65')]} end of tocinfo --> @@ -229,7 +171,7 @@ MathJax.Hub.Config({ - Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks + Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning
    @@ -303,44 +245,6 @@ MathJax.Hub.Config({
  • Which activation function should we use?
  • A top-down perspective on Neural networks
  • Limitations of supervised learning with deep networks
    • -
  • Convolutional Neural Networks (recognizing images)
    • -
  • Regular NNs don’t scale well to full images
    • -
  • 3D volumes of neurons
    • -
  • Layers used to build CNNs
    • -
  • Transforming images
    • -
  • CNNs in brief
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
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  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Using TensorFlow backend
    • -
  • Train the model
    • -
  • Visualizing the results
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • Fun links
    • -
  • Applications: solving ordinary differential equations with Neural Networks
    • -
  • Trial solution
    • -
  • More details
    • -
  • Reformulating the problem
    • -
  • Estimating errors
    • -
  • Creating a simple Deep Neural Net
    • -
  • Setting up the code, feed forward part
    • -
  • Backpropagation
    • -
  • Gradient Descent
    • -
  • More on GD and cost function
    • -
  • An implementation of a Deep Neural Network
    • -
  • The final parts of the code
    • -
  • And adding Back propagation
    • -
  • Solving the ODE
    • -
  • Using neural network
    • -
  • Using a deep neural network
    • -
  • Wrapping it up
    • @@ -401,7 +305,7 @@ which in case of the simply binary model reduces to having \( i=j \).
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs035.html b/doc/pub/NeuralNet/html/._NeuralNet-bs035.html index b108344a5..54e42a9fd 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs035.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs035.html @@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source - + -Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks +Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning @@ -141,65 +141,7 @@ Automatically generated HTML file from DocOnce source ('Limitations of supervised learning with deep networks', 2, None, - '___sec65'), - ('Convolutional Neural Networks (recognizing images)', - 2, - None, - '___sec66'), - ('Regular NNs don’t scale well to full images', - 2, - None, - '___sec67'), - ('3D volumes of neurons', 2, None, '___sec68'), - ('Layers used to build CNNs', 2, None, '___sec69'), - ('Transforming images', 2, None, '___sec70'), - ('CNNs in brief', 2, None, '___sec71'), - ('CNNs in more detail, building convolutional neural networks in ' - 'Tensorflow and Keras', - 2, - None, - '___sec72'), - ('Setting it up', 2, None, '___sec73'), - ('The MNIST dataset again', 2, None, '___sec74'), - ('Strong correlations', 2, None, '___sec75'), - ('Layers of a CNN', 2, None, '___sec76'), - ('Systematic reduction', 2, None, '___sec77'), - ('Prerequisites: Collect and pre-process data', - 2, - None, - '___sec78'), - ('Importing Keras and Tensorflow', 2, None, '___sec79'), - ('Using TensorFlow backend', 2, None, '___sec80'), - ('Train the model', 2, None, '___sec81'), - ('Visualizing the results', 2, None, '___sec82'), - ('Running with Keras', 2, None, '___sec83'), - ('Final part', 2, None, '___sec84'), - ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86'), - ('Applications: solving ordinary differential equations with ' - 'Neural Networks', - 2, - None, - '___sec87'), - ('Trial solution', 2, None, '___sec88'), - ('More details', 2, None, '___sec89'), - ('Reformulating the problem', 2, None, '___sec90'), - ('Estimating errors', 2, None, '___sec91'), - ('Creating a simple Deep Neural Net', 2, None, '___sec92'), - ('Setting up the code, feed forward part', 2, None, '___sec93'), - ('Backpropagation', 2, None, '___sec94'), - ('Gradient Descent', 2, None, '___sec95'), - ('More on GD and cost function', 2, None, '___sec96'), - ('An implementation of a Deep Neural Network', - 2, - None, - '___sec97'), - ('The final parts of the code', 2, None, '___sec98'), - ('And adding Back propagation', 2, None, '___sec99'), - ('Solving the ODE', 2, None, '___sec100'), - ('Using neural network', 2, None, '___sec101'), - ('Using a deep neural network', 2, None, '___sec102'), - ('Wrapping it up', 2, None, '___sec103')]} + '___sec65')]} end of tocinfo --> @@ -229,7 +171,7 @@ MathJax.Hub.Config({ - Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks + Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning
    @@ -303,44 +245,6 @@ MathJax.Hub.Config({
  • Which activation function should we use?
  • A top-down perspective on Neural networks
  • Limitations of supervised learning with deep networks
    • -
  • Convolutional Neural Networks (recognizing images)
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  • Regular NNs don’t scale well to full images
    • -
  • 3D volumes of neurons
    • -
  • Layers used to build CNNs
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  • Transforming images
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  • CNNs in brief
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  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
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  • Setting it up
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  • The MNIST dataset again
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  • Strong correlations
    • -
  • Layers of a CNN
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  • Systematic reduction
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  • Prerequisites: Collect and pre-process data
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  • Importing Keras and Tensorflow
    • -
  • Using TensorFlow backend
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  • Train the model
    • -
  • Visualizing the results
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  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • Fun links
    • -
  • Applications: solving ordinary differential equations with Neural Networks
    • -
  • Trial solution
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  • More details
    • -
  • Reformulating the problem
    • -
  • Estimating errors
    • -
  • Creating a simple Deep Neural Net
    • -
  • Setting up the code, feed forward part
    • -
  • Backpropagation
    • -
  • Gradient Descent
    • -
  • More on GD and cost function
    • -
  • An implementation of a Deep Neural Network
    • -
  • The final parts of the code
    • -
  • And adding Back propagation
    • -
  • Solving the ODE
    • -
  • Using neural network
    • -
  • Using a deep neural network
    • -
  • Wrapping it up
    • @@ -395,7 +299,7 @@ One can identify a set of key steps when using neural networks to solve supervis
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs036.html b/doc/pub/NeuralNet/html/._NeuralNet-bs036.html index 8c6088a45..e0d2d92ea 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs036.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs036.html @@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source - + -Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks +Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning @@ -141,65 +141,7 @@ Automatically generated HTML file from DocOnce source ('Limitations of supervised learning with deep networks', 2, None, - '___sec65'), - ('Convolutional Neural Networks (recognizing images)', - 2, - None, - '___sec66'), - ('Regular NNs don’t scale well to full images', - 2, - None, - '___sec67'), - ('3D volumes of neurons', 2, None, '___sec68'), - ('Layers used to build CNNs', 2, None, '___sec69'), - ('Transforming images', 2, None, '___sec70'), - ('CNNs in brief', 2, None, '___sec71'), - ('CNNs in more detail, building convolutional neural networks in ' - 'Tensorflow and Keras', - 2, - None, - '___sec72'), - ('Setting it up', 2, None, '___sec73'), - ('The MNIST dataset again', 2, None, '___sec74'), - ('Strong correlations', 2, None, '___sec75'), - ('Layers of a CNN', 2, None, '___sec76'), - ('Systematic reduction', 2, None, '___sec77'), - ('Prerequisites: Collect and pre-process data', - 2, - None, - '___sec78'), - ('Importing Keras and Tensorflow', 2, None, '___sec79'), - ('Using TensorFlow backend', 2, None, '___sec80'), - ('Train the model', 2, None, '___sec81'), - ('Visualizing the results', 2, None, '___sec82'), - ('Running with Keras', 2, None, '___sec83'), - ('Final part', 2, None, '___sec84'), - ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86'), - ('Applications: solving ordinary differential equations with ' - 'Neural Networks', - 2, - None, - '___sec87'), - ('Trial solution', 2, None, '___sec88'), - ('More details', 2, None, '___sec89'), - ('Reformulating the problem', 2, None, '___sec90'), - ('Estimating errors', 2, None, '___sec91'), - ('Creating a simple Deep Neural Net', 2, None, '___sec92'), - ('Setting up the code, feed forward part', 2, None, '___sec93'), - ('Backpropagation', 2, None, '___sec94'), - ('Gradient Descent', 2, None, '___sec95'), - ('More on GD and cost function', 2, None, '___sec96'), - ('An implementation of a Deep Neural Network', - 2, - None, - '___sec97'), - ('The final parts of the code', 2, None, '___sec98'), - ('And adding Back propagation', 2, None, '___sec99'), - ('Solving the ODE', 2, None, '___sec100'), - ('Using neural network', 2, None, '___sec101'), - ('Using a deep neural network', 2, None, '___sec102'), - ('Wrapping it up', 2, None, '___sec103')]} + '___sec65')]} end of tocinfo --> @@ -229,7 +171,7 @@ MathJax.Hub.Config({ - Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks + Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning
    @@ -303,44 +245,6 @@ MathJax.Hub.Config({
  • Which activation function should we use?
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  • Limitations of supervised learning with deep networks
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  • Convolutional Neural Networks (recognizing images)
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  • Regular NNs don’t scale well to full images
    • -
  • 3D volumes of neurons
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  • Layers used to build CNNs
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  • Transforming images
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  • CNNs in brief
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  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
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  • The MNIST dataset again
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  • Strong correlations
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  • Layers of a CNN
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  • Systematic reduction
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  • Using TensorFlow backend
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  • Visualizing the results
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  • Running with Keras
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  • Final part
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  • Final visualization
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  • Fun links
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  • Trial solution
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  • More details
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  • Estimating errors
    • -
  • Creating a simple Deep Neural Net
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  • Setting up the code, feed forward part
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  • Backpropagation
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  • Gradient Descent
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  • More on GD and cost function
    • -
  • An implementation of a Deep Neural Network
    • -
  • The final parts of the code
    • -
  • And adding Back propagation
    • -
  • Solving the ODE
    • -
  • Using neural network
    • -
  • Using a deep neural network
    • -
  • Wrapping it up
    • @@ -474,7 +378,7 @@ plt.show()
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs037.html b/doc/pub/NeuralNet/html/._NeuralNet-bs037.html index d3c9992e5..8250e13cc 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs037.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs037.html @@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source - + -Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks +Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning @@ -141,65 +141,7 @@ Automatically generated HTML file from DocOnce source ('Limitations of supervised learning with deep networks', 2, None, - '___sec65'), - ('Convolutional Neural Networks (recognizing images)', - 2, - None, - '___sec66'), - ('Regular NNs don’t scale well to full images', - 2, - None, - '___sec67'), - ('3D volumes of neurons', 2, None, '___sec68'), - ('Layers used to build CNNs', 2, None, '___sec69'), - ('Transforming images', 2, None, '___sec70'), - ('CNNs in brief', 2, None, '___sec71'), - ('CNNs in more detail, building convolutional neural networks in ' - 'Tensorflow and Keras', - 2, - None, - '___sec72'), - ('Setting it up', 2, None, '___sec73'), - ('The MNIST dataset again', 2, None, '___sec74'), - ('Strong correlations', 2, None, '___sec75'), - ('Layers of a CNN', 2, None, '___sec76'), - ('Systematic reduction', 2, None, '___sec77'), - ('Prerequisites: Collect and pre-process data', - 2, - None, - '___sec78'), - ('Importing Keras and Tensorflow', 2, None, '___sec79'), - ('Using TensorFlow backend', 2, None, '___sec80'), - ('Train the model', 2, None, '___sec81'), - ('Visualizing the results', 2, None, '___sec82'), - ('Running with Keras', 2, None, '___sec83'), - ('Final part', 2, None, '___sec84'), - ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86'), - ('Applications: solving ordinary differential equations with ' - 'Neural Networks', - 2, - None, - '___sec87'), - ('Trial solution', 2, None, '___sec88'), - ('More details', 2, None, '___sec89'), - ('Reformulating the problem', 2, None, '___sec90'), - ('Estimating errors', 2, None, '___sec91'), - ('Creating a simple Deep Neural Net', 2, None, '___sec92'), - ('Setting up the code, feed forward part', 2, None, '___sec93'), - ('Backpropagation', 2, None, '___sec94'), - ('Gradient Descent', 2, None, '___sec95'), - ('More on GD and cost function', 2, None, '___sec96'), - ('An implementation of a Deep Neural Network', - 2, - None, - '___sec97'), - ('The final parts of the code', 2, None, '___sec98'), - ('And adding Back propagation', 2, None, '___sec99'), - ('Solving the ODE', 2, None, '___sec100'), - ('Using neural network', 2, None, '___sec101'), - ('Using a deep neural network', 2, None, '___sec102'), - ('Wrapping it up', 2, None, '___sec103')]} + '___sec65')]} end of tocinfo --> @@ -229,7 +171,7 @@ MathJax.Hub.Config({ - Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks + Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning
    @@ -303,44 +245,6 @@ MathJax.Hub.Config({
  • Which activation function should we use?
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  • Limitations of supervised learning with deep networks
    • -
  • Convolutional Neural Networks (recognizing images)
    • -
  • Regular NNs don’t scale well to full images
    • -
  • 3D volumes of neurons
    • -
  • Layers used to build CNNs
    • -
  • Transforming images
    • -
  • CNNs in brief
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
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  • Setting it up
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  • Strong correlations
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  • Layers of a CNN
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  • Systematic reduction
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  • Prerequisites: Collect and pre-process data
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  • Importing Keras and Tensorflow
    • -
  • Using TensorFlow backend
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  • Train the model
    • -
  • Visualizing the results
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • Fun links
    • -
  • Applications: solving ordinary differential equations with Neural Networks
    • -
  • Trial solution
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  • More details
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  • Reformulating the problem
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  • Estimating errors
    • -
  • Creating a simple Deep Neural Net
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  • Setting up the code, feed forward part
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  • Backpropagation
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  • Gradient Descent
    • -
  • More on GD and cost function
    • -
  • An implementation of a Deep Neural Network
    • -
  • The final parts of the code
    • -
  • And adding Back propagation
    • -
  • Solving the ODE
    • -
  • Using neural network
    • -
  • Using a deep neural network
    • -
  • Wrapping it up
    • @@ -428,7 +332,7 @@ X_train, X_test, Y_train, Y_test = train_tes
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs038.html b/doc/pub/NeuralNet/html/._NeuralNet-bs038.html index e05ec4ae3..ee766804c 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs038.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs038.html @@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source - + -Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks +Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning @@ -141,65 +141,7 @@ Automatically generated HTML file from DocOnce source ('Limitations of supervised learning with deep networks', 2, None, - '___sec65'), - ('Convolutional Neural Networks (recognizing images)', - 2, - None, - '___sec66'), - ('Regular NNs don’t scale well to full images', - 2, - None, - '___sec67'), - ('3D volumes of neurons', 2, None, '___sec68'), - ('Layers used to build CNNs', 2, None, '___sec69'), - ('Transforming images', 2, None, '___sec70'), - ('CNNs in brief', 2, None, '___sec71'), - ('CNNs in more detail, building convolutional neural networks in ' - 'Tensorflow and Keras', - 2, - None, - '___sec72'), - ('Setting it up', 2, None, '___sec73'), - ('The MNIST dataset again', 2, None, '___sec74'), - ('Strong correlations', 2, None, '___sec75'), - ('Layers of a CNN', 2, None, '___sec76'), - ('Systematic reduction', 2, None, '___sec77'), - ('Prerequisites: Collect and pre-process data', - 2, - None, - '___sec78'), - ('Importing Keras and Tensorflow', 2, None, '___sec79'), - ('Using TensorFlow backend', 2, None, '___sec80'), - ('Train the model', 2, None, '___sec81'), - ('Visualizing the results', 2, None, '___sec82'), - ('Running with Keras', 2, None, '___sec83'), - ('Final part', 2, None, '___sec84'), - ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86'), - ('Applications: solving ordinary differential equations with ' - 'Neural Networks', - 2, - None, - '___sec87'), - ('Trial solution', 2, None, '___sec88'), - ('More details', 2, None, '___sec89'), - ('Reformulating the problem', 2, None, '___sec90'), - ('Estimating errors', 2, None, '___sec91'), - ('Creating a simple Deep Neural Net', 2, None, '___sec92'), - ('Setting up the code, feed forward part', 2, None, '___sec93'), - ('Backpropagation', 2, None, '___sec94'), - ('Gradient Descent', 2, None, '___sec95'), - ('More on GD and cost function', 2, None, '___sec96'), - ('An implementation of a Deep Neural Network', - 2, - None, - '___sec97'), - ('The final parts of the code', 2, None, '___sec98'), - ('And adding Back propagation', 2, None, '___sec99'), - ('Solving the ODE', 2, None, '___sec100'), - ('Using neural network', 2, None, '___sec101'), - ('Using a deep neural network', 2, None, '___sec102'), - ('Wrapping it up', 2, None, '___sec103')]} + '___sec65')]} end of tocinfo --> @@ -229,7 +171,7 @@ MathJax.Hub.Config({ - Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks + Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning
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  • Which activation function should we use?
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    • -
  • 3D volumes of neurons
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  • Using TensorFlow backend
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  • Visualizing the results
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  • Running with Keras
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  • Final part
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  • Final visualization
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  • Fun links
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  • Gradient Descent
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  • More on GD and cost function
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  • An implementation of a Deep Neural Network
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  • The final parts of the code
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  • And adding Back propagation
    • -
  • Solving the ODE
    • -
  • Using neural network
    • -
  • Using a deep neural network
    • -
  • Wrapping it up
    • @@ -425,7 +329,7 @@ which is inspired by probability theory (see logistic regression) and was most c
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs039.html b/doc/pub/NeuralNet/html/._NeuralNet-bs039.html index 765e83e84..cd55dff4e 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs039.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs039.html @@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source - + -Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks +Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning @@ -141,65 +141,7 @@ Automatically generated HTML file from DocOnce source ('Limitations of supervised learning with deep networks', 2, None, - '___sec65'), - ('Convolutional Neural Networks (recognizing images)', - 2, - None, - '___sec66'), - ('Regular NNs don’t scale well to full images', - 2, - None, - '___sec67'), - ('3D volumes of neurons', 2, None, '___sec68'), - ('Layers used to build CNNs', 2, None, '___sec69'), - ('Transforming images', 2, None, '___sec70'), - ('CNNs in brief', 2, None, '___sec71'), - ('CNNs in more detail, building convolutional neural networks in ' - 'Tensorflow and Keras', - 2, - None, - '___sec72'), - ('Setting it up', 2, None, '___sec73'), - ('The MNIST dataset again', 2, None, '___sec74'), - ('Strong correlations', 2, None, '___sec75'), - ('Layers of a CNN', 2, None, '___sec76'), - ('Systematic reduction', 2, None, '___sec77'), - ('Prerequisites: Collect and pre-process data', - 2, - None, - '___sec78'), - ('Importing Keras and Tensorflow', 2, None, '___sec79'), - ('Using TensorFlow backend', 2, None, '___sec80'), - ('Train the model', 2, None, '___sec81'), - ('Visualizing the results', 2, None, '___sec82'), - ('Running with Keras', 2, None, '___sec83'), - ('Final part', 2, None, '___sec84'), - ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86'), - ('Applications: solving ordinary differential equations with ' - 'Neural Networks', - 2, - None, - '___sec87'), - ('Trial solution', 2, None, '___sec88'), - ('More details', 2, None, '___sec89'), - ('Reformulating the problem', 2, None, '___sec90'), - ('Estimating errors', 2, None, '___sec91'), - ('Creating a simple Deep Neural Net', 2, None, '___sec92'), - ('Setting up the code, feed forward part', 2, None, '___sec93'), - ('Backpropagation', 2, None, '___sec94'), - ('Gradient Descent', 2, None, '___sec95'), - ('More on GD and cost function', 2, None, '___sec96'), - ('An implementation of a Deep Neural Network', - 2, - None, - '___sec97'), - ('The final parts of the code', 2, None, '___sec98'), - ('And adding Back propagation', 2, None, '___sec99'), - ('Solving the ODE', 2, None, '___sec100'), - ('Using neural network', 2, None, '___sec101'), - ('Using a deep neural network', 2, None, '___sec102'), - ('Wrapping it up', 2, None, '___sec103')]} + '___sec65')]} end of tocinfo --> @@ -229,7 +171,7 @@ MathJax.Hub.Config({ - Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks + Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning
    @@ -303,44 +245,6 @@ MathJax.Hub.Config({
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  • Regular NNs don’t scale well to full images
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  • 3D volumes of neurons
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  • Transforming images
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  • CNNs in brief
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  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
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  • Setting it up
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  • The MNIST dataset again
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  • Strong correlations
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  • Layers of a CNN
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  • Systematic reduction
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  • More on GD and cost function
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  • An implementation of a Deep Neural Network
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  • The final parts of the code
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  • And adding Back propagation
    • -
  • Solving the ODE
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  • Using neural network
    • -
  • Using a deep neural network
    • -
  • Wrapping it up
    • @@ -424,7 +328,7 @@ weights to the output layer.
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs040.html b/doc/pub/NeuralNet/html/._NeuralNet-bs040.html index 4201a313d..bbb568553 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs040.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs040.html @@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source - + -Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks +Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning @@ -141,65 +141,7 @@ Automatically generated HTML file from DocOnce source ('Limitations of supervised learning with deep networks', 2, None, - '___sec65'), - ('Convolutional Neural Networks (recognizing images)', - 2, - None, - '___sec66'), - ('Regular NNs don’t scale well to full images', - 2, - None, - '___sec67'), - ('3D volumes of neurons', 2, None, '___sec68'), - ('Layers used to build CNNs', 2, None, '___sec69'), - ('Transforming images', 2, None, '___sec70'), - ('CNNs in brief', 2, None, '___sec71'), - ('CNNs in more detail, building convolutional neural networks in ' - 'Tensorflow and Keras', - 2, - None, - '___sec72'), - ('Setting it up', 2, None, '___sec73'), - ('The MNIST dataset again', 2, None, '___sec74'), - ('Strong correlations', 2, None, '___sec75'), - ('Layers of a CNN', 2, None, '___sec76'), - ('Systematic reduction', 2, None, '___sec77'), - ('Prerequisites: Collect and pre-process data', - 2, - None, - '___sec78'), - ('Importing Keras and Tensorflow', 2, None, '___sec79'), - ('Using TensorFlow backend', 2, None, '___sec80'), - ('Train the model', 2, None, '___sec81'), - ('Visualizing the results', 2, None, '___sec82'), - ('Running with Keras', 2, None, '___sec83'), - ('Final part', 2, None, '___sec84'), - ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86'), - ('Applications: solving ordinary differential equations with ' - 'Neural Networks', - 2, - None, - '___sec87'), - ('Trial solution', 2, None, '___sec88'), - ('More details', 2, None, '___sec89'), - ('Reformulating the problem', 2, None, '___sec90'), - ('Estimating errors', 2, None, '___sec91'), - ('Creating a simple Deep Neural Net', 2, None, '___sec92'), - ('Setting up the code, feed forward part', 2, None, '___sec93'), - ('Backpropagation', 2, None, '___sec94'), - ('Gradient Descent', 2, None, '___sec95'), - ('More on GD and cost function', 2, None, '___sec96'), - ('An implementation of a Deep Neural Network', - 2, - None, - '___sec97'), - ('The final parts of the code', 2, None, '___sec98'), - ('And adding Back propagation', 2, None, '___sec99'), - ('Solving the ODE', 2, None, '___sec100'), - ('Using neural network', 2, None, '___sec101'), - ('Using a deep neural network', 2, None, '___sec102'), - ('Wrapping it up', 2, None, '___sec103')]} + '___sec65')]} end of tocinfo --> @@ -229,7 +171,7 @@ MathJax.Hub.Config({ - Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks + Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning
    @@ -303,44 +245,6 @@ MathJax.Hub.Config({
  • Which activation function should we use?
  • A top-down perspective on Neural networks
  • Limitations of supervised learning with deep networks
    • -
  • Convolutional Neural Networks (recognizing images)
    • -
  • Regular NNs don’t scale well to full images
    • -
  • 3D volumes of neurons
    • -
  • Layers used to build CNNs
    • -
  • Transforming images
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  • CNNs in brief
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
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  • Systematic reduction
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  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Using TensorFlow backend
    • -
  • Train the model
    • -
  • Visualizing the results
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • Fun links
    • -
  • Applications: solving ordinary differential equations with Neural Networks
    • -
  • Trial solution
    • -
  • More details
    • -
  • Reformulating the problem
    • -
  • Estimating errors
    • -
  • Creating a simple Deep Neural Net
    • -
  • Setting up the code, feed forward part
    • -
  • Backpropagation
    • -
  • Gradient Descent
    • -
  • More on GD and cost function
    • -
  • An implementation of a Deep Neural Network
    • -
  • The final parts of the code
    • -
  • And adding Back propagation
    • -
  • Solving the ODE
    • -
  • Using neural network
    • -
  • Using a deep neural network
    • -
  • Wrapping it up
    • @@ -415,7 +319,7 @@ output_bias = np49
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs041.html b/doc/pub/NeuralNet/html/._NeuralNet-bs041.html index bda78a044..c0c97576e 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs041.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs041.html @@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source - + -Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks +Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning @@ -141,65 +141,7 @@ Automatically generated HTML file from DocOnce source ('Limitations of supervised learning with deep networks', 2, None, - '___sec65'), - ('Convolutional Neural Networks (recognizing images)', - 2, - None, - '___sec66'), - ('Regular NNs don’t scale well to full images', - 2, - None, - '___sec67'), - ('3D volumes of neurons', 2, None, '___sec68'), - ('Layers used to build CNNs', 2, None, '___sec69'), - ('Transforming images', 2, None, '___sec70'), - ('CNNs in brief', 2, None, '___sec71'), - ('CNNs in more detail, building convolutional neural networks in ' - 'Tensorflow and Keras', - 2, - None, - '___sec72'), - ('Setting it up', 2, None, '___sec73'), - ('The MNIST dataset again', 2, None, '___sec74'), - ('Strong correlations', 2, None, '___sec75'), - ('Layers of a CNN', 2, None, '___sec76'), - ('Systematic reduction', 2, None, '___sec77'), - ('Prerequisites: Collect and pre-process data', - 2, - None, - '___sec78'), - ('Importing Keras and Tensorflow', 2, None, '___sec79'), - ('Using TensorFlow backend', 2, None, '___sec80'), - ('Train the model', 2, None, '___sec81'), - ('Visualizing the results', 2, None, '___sec82'), - ('Running with Keras', 2, None, '___sec83'), - ('Final part', 2, None, '___sec84'), - ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86'), - ('Applications: solving ordinary differential equations with ' - 'Neural Networks', - 2, - None, - '___sec87'), - ('Trial solution', 2, None, '___sec88'), - ('More details', 2, None, '___sec89'), - ('Reformulating the problem', 2, None, '___sec90'), - ('Estimating errors', 2, None, '___sec91'), - ('Creating a simple Deep Neural Net', 2, None, '___sec92'), - ('Setting up the code, feed forward part', 2, None, '___sec93'), - ('Backpropagation', 2, None, '___sec94'), - ('Gradient Descent', 2, None, '___sec95'), - ('More on GD and cost function', 2, None, '___sec96'), - ('An implementation of a Deep Neural Network', - 2, - None, - '___sec97'), - ('The final parts of the code', 2, None, '___sec98'), - ('And adding Back propagation', 2, None, '___sec99'), - ('Solving the ODE', 2, None, '___sec100'), - ('Using neural network', 2, None, '___sec101'), - ('Using a deep neural network', 2, None, '___sec102'), - ('Wrapping it up', 2, None, '___sec103')]} + '___sec65')]} end of tocinfo --> @@ -229,7 +171,7 @@ MathJax.Hub.Config({ - Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks + Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning
    @@ -303,44 +245,6 @@ MathJax.Hub.Config({
  • Which activation function should we use?
  • A top-down perspective on Neural networks
  • Limitations of supervised learning with deep networks
    • -
  • Convolutional Neural Networks (recognizing images)
    • -
  • Regular NNs don’t scale well to full images
    • -
  • 3D volumes of neurons
    • -
  • Layers used to build CNNs
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  • Transforming images
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  • CNNs in brief
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
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  • Setting it up
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  • The MNIST dataset again
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  • Strong correlations
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  • Layers of a CNN
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  • Systematic reduction
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  • Prerequisites: Collect and pre-process data
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  • Importing Keras and Tensorflow
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  • Using TensorFlow backend
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  • Train the model
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  • Visualizing the results
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  • Running with Keras
    • -
  • Final part
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  • Final visualization
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  • Fun links
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  • Applications: solving ordinary differential equations with Neural Networks
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  • Trial solution
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  • More details
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  • Reformulating the problem
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  • Estimating errors
    • -
  • Creating a simple Deep Neural Net
    • -
  • Setting up the code, feed forward part
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  • Backpropagation
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  • Gradient Descent
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  • More on GD and cost function
    • -
  • An implementation of a Deep Neural Network
    • -
  • The final parts of the code
    • -
  • And adding Back propagation
    • -
  • Solving the ODE
    • -
  • Using neural network
    • -
  • Using a deep neural network
    • -
  • Wrapping it up
    • @@ -406,7 +310,7 @@ $$ a_{j}^{L} = \frac{\exp{(z_j^{L})}}
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs042.html b/doc/pub/NeuralNet/html/._NeuralNet-bs042.html index 528eece01..89e3c5cc6 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs042.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs042.html @@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source - + -Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks +Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning @@ -141,65 +141,7 @@ Automatically generated HTML file from DocOnce source ('Limitations of supervised learning with deep networks', 2, None, - '___sec65'), - ('Convolutional Neural Networks (recognizing images)', - 2, - None, - '___sec66'), - ('Regular NNs don’t scale well to full images', - 2, - None, - '___sec67'), - ('3D volumes of neurons', 2, None, '___sec68'), - ('Layers used to build CNNs', 2, None, '___sec69'), - ('Transforming images', 2, None, '___sec70'), - ('CNNs in brief', 2, None, '___sec71'), - ('CNNs in more detail, building convolutional neural networks in ' - 'Tensorflow and Keras', - 2, - None, - '___sec72'), - ('Setting it up', 2, None, '___sec73'), - ('The MNIST dataset again', 2, None, '___sec74'), - ('Strong correlations', 2, None, '___sec75'), - ('Layers of a CNN', 2, None, '___sec76'), - ('Systematic reduction', 2, None, '___sec77'), - ('Prerequisites: Collect and pre-process data', - 2, - None, - '___sec78'), - ('Importing Keras and Tensorflow', 2, None, '___sec79'), - ('Using TensorFlow backend', 2, None, '___sec80'), - ('Train the model', 2, None, '___sec81'), - ('Visualizing the results', 2, None, '___sec82'), - ('Running with Keras', 2, None, '___sec83'), - ('Final part', 2, None, '___sec84'), - ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86'), - ('Applications: solving ordinary differential equations with ' - 'Neural Networks', - 2, - None, - '___sec87'), - ('Trial solution', 2, None, '___sec88'), - ('More details', 2, None, '___sec89'), - ('Reformulating the problem', 2, None, '___sec90'), - ('Estimating errors', 2, None, '___sec91'), - ('Creating a simple Deep Neural Net', 2, None, '___sec92'), - ('Setting up the code, feed forward part', 2, None, '___sec93'), - ('Backpropagation', 2, None, '___sec94'), - ('Gradient Descent', 2, None, '___sec95'), - ('More on GD and cost function', 2, None, '___sec96'), - ('An implementation of a Deep Neural Network', - 2, - None, - '___sec97'), - ('The final parts of the code', 2, None, '___sec98'), - ('And adding Back propagation', 2, None, '___sec99'), - ('Solving the ODE', 2, None, '___sec100'), - ('Using neural network', 2, None, '___sec101'), - ('Using a deep neural network', 2, None, '___sec102'), - ('Wrapping it up', 2, None, '___sec103')]} + '___sec65')]} end of tocinfo --> @@ -229,7 +171,7 @@ MathJax.Hub.Config({ - Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks + Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning
    @@ -303,44 +245,6 @@ MathJax.Hub.Config({
  • Which activation function should we use?
  • A top-down perspective on Neural networks
  • Limitations of supervised learning with deep networks
    • -
  • Convolutional Neural Networks (recognizing images)
    • -
  • Regular NNs don’t scale well to full images
    • -
  • 3D volumes of neurons
    • -
  • Layers used to build CNNs
    • -
  • Transforming images
    • -
  • CNNs in brief
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
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  • Setting it up
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  • The MNIST dataset again
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  • Strong correlations
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  • Layers of a CNN
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  • Systematic reduction
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  • Prerequisites: Collect and pre-process data
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  • Importing Keras and Tensorflow
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  • Using TensorFlow backend
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  • Train the model
    • -
  • Visualizing the results
    • -
  • Running with Keras
    • -
  • Final part
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  • Final visualization
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  • Fun links
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  • Applications: solving ordinary differential equations with Neural Networks
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  • Trial solution
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  • More details
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  • Reformulating the problem
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  • Estimating errors
    • -
  • Creating a simple Deep Neural Net
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  • Setting up the code, feed forward part
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  • Backpropagation
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  • Gradient Descent
    • -
  • More on GD and cost function
    • -
  • An implementation of a Deep Neural Network
    • -
  • The final parts of the code
    • -
  • And adding Back propagation
    • -
  • Solving the ODE
    • -
  • Using neural network
    • -
  • Using a deep neural network
    • -
  • Wrapping it up
    • @@ -454,7 +358,7 @@ predictions = predict(X_train)
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs043.html b/doc/pub/NeuralNet/html/._NeuralNet-bs043.html index 52b8bad03..fe03404f8 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs043.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs043.html @@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source - + -Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks +Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning @@ -141,65 +141,7 @@ Automatically generated HTML file from DocOnce source ('Limitations of supervised learning with deep networks', 2, None, - '___sec65'), - ('Convolutional Neural Networks (recognizing images)', - 2, - None, - '___sec66'), - ('Regular NNs don’t scale well to full images', - 2, - None, - '___sec67'), - ('3D volumes of neurons', 2, None, '___sec68'), - ('Layers used to build CNNs', 2, None, '___sec69'), - ('Transforming images', 2, None, '___sec70'), - ('CNNs in brief', 2, None, '___sec71'), - ('CNNs in more detail, building convolutional neural networks in ' - 'Tensorflow and Keras', - 2, - None, - '___sec72'), - ('Setting it up', 2, None, '___sec73'), - ('The MNIST dataset again', 2, None, '___sec74'), - ('Strong correlations', 2, None, '___sec75'), - ('Layers of a CNN', 2, None, '___sec76'), - ('Systematic reduction', 2, None, '___sec77'), - ('Prerequisites: Collect and pre-process data', - 2, - None, - '___sec78'), - ('Importing Keras and Tensorflow', 2, None, '___sec79'), - ('Using TensorFlow backend', 2, None, '___sec80'), - ('Train the model', 2, None, '___sec81'), - ('Visualizing the results', 2, None, '___sec82'), - ('Running with Keras', 2, None, '___sec83'), - ('Final part', 2, None, '___sec84'), - ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86'), - ('Applications: solving ordinary differential equations with ' - 'Neural Networks', - 2, - None, - '___sec87'), - ('Trial solution', 2, None, '___sec88'), - ('More details', 2, None, '___sec89'), - ('Reformulating the problem', 2, None, '___sec90'), - ('Estimating errors', 2, None, '___sec91'), - ('Creating a simple Deep Neural Net', 2, None, '___sec92'), - ('Setting up the code, feed forward part', 2, None, '___sec93'), - ('Backpropagation', 2, None, '___sec94'), - ('Gradient Descent', 2, None, '___sec95'), - ('More on GD and cost function', 2, None, '___sec96'), - ('An implementation of a Deep Neural Network', - 2, - None, - '___sec97'), - ('The final parts of the code', 2, None, '___sec98'), - ('And adding Back propagation', 2, None, '___sec99'), - ('Solving the ODE', 2, None, '___sec100'), - ('Using neural network', 2, None, '___sec101'), - ('Using a deep neural network', 2, None, '___sec102'), - ('Wrapping it up', 2, None, '___sec103')]} + '___sec65')]} end of tocinfo --> @@ -229,7 +171,7 @@ MathJax.Hub.Config({ - Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks + Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning
    @@ -303,44 +245,6 @@ MathJax.Hub.Config({
  • Which activation function should we use?
  • A top-down perspective on Neural networks
  • Limitations of supervised learning with deep networks
    • -
  • Convolutional Neural Networks (recognizing images)
    • -
  • Regular NNs don’t scale well to full images
    • -
  • 3D volumes of neurons
    • -
  • Layers used to build CNNs
    • -
  • Transforming images
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  • CNNs in brief
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
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  • Setting it up
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  • The MNIST dataset again
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  • Strong correlations
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  • Layers of a CNN
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  • Systematic reduction
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  • Prerequisites: Collect and pre-process data
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  • Importing Keras and Tensorflow
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  • Using TensorFlow backend
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  • Train the model
    • -
  • Visualizing the results
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  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
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  • Fun links
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  • Trial solution
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  • Estimating errors
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  • Creating a simple Deep Neural Net
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  • Setting up the code, feed forward part
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  • Backpropagation
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  • Gradient Descent
    • -
  • More on GD and cost function
    • -
  • An implementation of a Deep Neural Network
    • -
  • The final parts of the code
    • -
  • And adding Back propagation
    • -
  • Solving the ODE
    • -
  • Using neural network
    • -
  • Using a deep neural network
    • -
  • Wrapping it up
    • @@ -410,7 +314,7 @@ you got the correct label. The probability of category \( c \) is given by the s
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs044.html b/doc/pub/NeuralNet/html/._NeuralNet-bs044.html index 33ef8fdf3..e67e0a3bd 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs044.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs044.html @@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source - + -Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks +Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning @@ -141,65 +141,7 @@ Automatically generated HTML file from DocOnce source ('Limitations of supervised learning with deep networks', 2, None, - '___sec65'), - ('Convolutional Neural Networks (recognizing images)', - 2, - None, - '___sec66'), - ('Regular NNs don’t scale well to full images', - 2, - None, - '___sec67'), - ('3D volumes of neurons', 2, None, '___sec68'), - ('Layers used to build CNNs', 2, None, '___sec69'), - ('Transforming images', 2, None, '___sec70'), - ('CNNs in brief', 2, None, '___sec71'), - ('CNNs in more detail, building convolutional neural networks in ' - 'Tensorflow and Keras', - 2, - None, - '___sec72'), - ('Setting it up', 2, None, '___sec73'), - ('The MNIST dataset again', 2, None, '___sec74'), - ('Strong correlations', 2, None, '___sec75'), - ('Layers of a CNN', 2, None, '___sec76'), - ('Systematic reduction', 2, None, '___sec77'), - ('Prerequisites: Collect and pre-process data', - 2, - None, - '___sec78'), - ('Importing Keras and Tensorflow', 2, None, '___sec79'), - ('Using TensorFlow backend', 2, None, '___sec80'), - ('Train the model', 2, None, '___sec81'), - ('Visualizing the results', 2, None, '___sec82'), - ('Running with Keras', 2, None, '___sec83'), - ('Final part', 2, None, '___sec84'), - ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86'), - ('Applications: solving ordinary differential equations with ' - 'Neural Networks', - 2, - None, - '___sec87'), - ('Trial solution', 2, None, '___sec88'), - ('More details', 2, None, '___sec89'), - ('Reformulating the problem', 2, None, '___sec90'), - ('Estimating errors', 2, None, '___sec91'), - ('Creating a simple Deep Neural Net', 2, None, '___sec92'), - ('Setting up the code, feed forward part', 2, None, '___sec93'), - ('Backpropagation', 2, None, '___sec94'), - ('Gradient Descent', 2, None, '___sec95'), - ('More on GD and cost function', 2, None, '___sec96'), - ('An implementation of a Deep Neural Network', - 2, - None, - '___sec97'), - ('The final parts of the code', 2, None, '___sec98'), - ('And adding Back propagation', 2, None, '___sec99'), - ('Solving the ODE', 2, None, '___sec100'), - ('Using neural network', 2, None, '___sec101'), - ('Using a deep neural network', 2, None, '___sec102'), - ('Wrapping it up', 2, None, '___sec103')]} + '___sec65')]} end of tocinfo --> @@ -229,7 +171,7 @@ MathJax.Hub.Config({ - Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks + Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning
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  • Which activation function should we use?
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  • Regular NNs don’t scale well to full images
    • -
  • 3D volumes of neurons
    • -
  • Layers used to build CNNs
    • -
  • Transforming images
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  • CNNs in brief
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  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
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  • Setting it up
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  • The MNIST dataset again
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  • Strong correlations
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  • Layers of a CNN
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  • Systematic reduction
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  • Visualizing the results
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  • Running with Keras
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  • Creating a simple Deep Neural Net
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  • Gradient Descent
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  • More on GD and cost function
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  • An implementation of a Deep Neural Network
    • -
  • The final parts of the code
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  • And adding Back propagation
    • -
  • Solving the ODE
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  • Using neural network
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  • Using a deep neural network
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    • @@ -419,7 +323,7 @@ The various optmization methods, with codes and algorithms, are discussed in o
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs045.html b/doc/pub/NeuralNet/html/._NeuralNet-bs045.html index bba1f13e8..e68408501 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs045.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs045.html @@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source - + -Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks +Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning @@ -141,65 +141,7 @@ Automatically generated HTML file from DocOnce source ('Limitations of supervised learning with deep networks', 2, None, - '___sec65'), - ('Convolutional Neural Networks (recognizing images)', - 2, - None, - '___sec66'), - ('Regular NNs don’t scale well to full images', - 2, - None, - '___sec67'), - ('3D volumes of neurons', 2, None, '___sec68'), - ('Layers used to build CNNs', 2, None, '___sec69'), - ('Transforming images', 2, None, '___sec70'), - ('CNNs in brief', 2, None, '___sec71'), - ('CNNs in more detail, building convolutional neural networks in ' - 'Tensorflow and Keras', - 2, - None, - '___sec72'), - ('Setting it up', 2, None, '___sec73'), - ('The MNIST dataset again', 2, None, '___sec74'), - ('Strong correlations', 2, None, '___sec75'), - ('Layers of a CNN', 2, None, '___sec76'), - ('Systematic reduction', 2, None, '___sec77'), - ('Prerequisites: Collect and pre-process data', - 2, - None, - '___sec78'), - ('Importing Keras and Tensorflow', 2, None, '___sec79'), - ('Using TensorFlow backend', 2, None, '___sec80'), - ('Train the model', 2, None, '___sec81'), - ('Visualizing the results', 2, None, '___sec82'), - ('Running with Keras', 2, None, '___sec83'), - ('Final part', 2, None, '___sec84'), - ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86'), - ('Applications: solving ordinary differential equations with ' - 'Neural Networks', - 2, - None, - '___sec87'), - ('Trial solution', 2, None, '___sec88'), - ('More details', 2, None, '___sec89'), - ('Reformulating the problem', 2, None, '___sec90'), - ('Estimating errors', 2, None, '___sec91'), - ('Creating a simple Deep Neural Net', 2, None, '___sec92'), - ('Setting up the code, feed forward part', 2, None, '___sec93'), - ('Backpropagation', 2, None, '___sec94'), - ('Gradient Descent', 2, None, '___sec95'), - ('More on GD and cost function', 2, None, '___sec96'), - ('An implementation of a Deep Neural Network', - 2, - None, - '___sec97'), - ('The final parts of the code', 2, None, '___sec98'), - ('And adding Back propagation', 2, None, '___sec99'), - ('Solving the ODE', 2, None, '___sec100'), - ('Using neural network', 2, None, '___sec101'), - ('Using a deep neural network', 2, None, '___sec102'), - ('Wrapping it up', 2, None, '___sec103')]} + '___sec65')]} end of tocinfo --> @@ -229,7 +171,7 @@ MathJax.Hub.Config({ - Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks + Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning
    @@ -303,44 +245,6 @@ MathJax.Hub.Config({
  • Which activation function should we use?
  • A top-down perspective on Neural networks
  • Limitations of supervised learning with deep networks
    • -
  • Convolutional Neural Networks (recognizing images)
    • -
  • Regular NNs don’t scale well to full images
    • -
  • 3D volumes of neurons
    • -
  • Layers used to build CNNs
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  • Transforming images
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  • CNNs in brief
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  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
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  • Setting it up
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  • The MNIST dataset again
    • -
  • Strong correlations
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  • Layers of a CNN
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  • Systematic reduction
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  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Using TensorFlow backend
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  • Train the model
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  • Visualizing the results
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  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • Fun links
    • -
  • Applications: solving ordinary differential equations with Neural Networks
    • -
  • Trial solution
    • -
  • More details
    • -
  • Reformulating the problem
    • -
  • Estimating errors
    • -
  • Creating a simple Deep Neural Net
    • -
  • Setting up the code, feed forward part
    • -
  • Backpropagation
    • -
  • Gradient Descent
    • -
  • More on GD and cost function
    • -
  • An implementation of a Deep Neural Network
    • -
  • The final parts of the code
    • -
  • And adding Back propagation
    • -
  • Solving the ODE
    • -
  • Using neural network
    • -
  • Using a deep neural network
    • -
  • Wrapping it up
    • @@ -412,7 +316,7 @@ calculate the gradient efficently.
  • 54
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs046.html b/doc/pub/NeuralNet/html/._NeuralNet-bs046.html index f26bbe9c5..2a5dca421 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs046.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs046.html @@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source - + -Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks +Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning @@ -141,65 +141,7 @@ Automatically generated HTML file from DocOnce source ('Limitations of supervised learning with deep networks', 2, None, - '___sec65'), - ('Convolutional Neural Networks (recognizing images)', - 2, - None, - '___sec66'), - ('Regular NNs don’t scale well to full images', - 2, - None, - '___sec67'), - ('3D volumes of neurons', 2, None, '___sec68'), - ('Layers used to build CNNs', 2, None, '___sec69'), - ('Transforming images', 2, None, '___sec70'), - ('CNNs in brief', 2, None, '___sec71'), - ('CNNs in more detail, building convolutional neural networks in ' - 'Tensorflow and Keras', - 2, - None, - '___sec72'), - ('Setting it up', 2, None, '___sec73'), - ('The MNIST dataset again', 2, None, '___sec74'), - ('Strong correlations', 2, None, '___sec75'), - ('Layers of a CNN', 2, None, '___sec76'), - ('Systematic reduction', 2, None, '___sec77'), - ('Prerequisites: Collect and pre-process data', - 2, - None, - '___sec78'), - ('Importing Keras and Tensorflow', 2, None, '___sec79'), - ('Using TensorFlow backend', 2, None, '___sec80'), - ('Train the model', 2, None, '___sec81'), - ('Visualizing the results', 2, None, '___sec82'), - ('Running with Keras', 2, None, '___sec83'), - ('Final part', 2, None, '___sec84'), - ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86'), - ('Applications: solving ordinary differential equations with ' - 'Neural Networks', - 2, - None, - '___sec87'), - ('Trial solution', 2, None, '___sec88'), - ('More details', 2, None, '___sec89'), - ('Reformulating the problem', 2, None, '___sec90'), - ('Estimating errors', 2, None, '___sec91'), - ('Creating a simple Deep Neural Net', 2, None, '___sec92'), - ('Setting up the code, feed forward part', 2, None, '___sec93'), - ('Backpropagation', 2, None, '___sec94'), - ('Gradient Descent', 2, None, '___sec95'), - ('More on GD and cost function', 2, None, '___sec96'), - ('An implementation of a Deep Neural Network', - 2, - None, - '___sec97'), - ('The final parts of the code', 2, None, '___sec98'), - ('And adding Back propagation', 2, None, '___sec99'), - ('Solving the ODE', 2, None, '___sec100'), - ('Using neural network', 2, None, '___sec101'), - ('Using a deep neural network', 2, None, '___sec102'), - ('Wrapping it up', 2, None, '___sec103')]} + '___sec65')]} end of tocinfo --> @@ -229,7 +171,7 @@ MathJax.Hub.Config({ - Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks + Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning
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  • Which activation function should we use?
  • A top-down perspective on Neural networks
  • Limitations of supervised learning with deep networks
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  • Convolutional Neural Networks (recognizing images)
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  • Regular NNs don’t scale well to full images
    • -
  • 3D volumes of neurons
    • -
  • Layers used to build CNNs
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  • Transforming images
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  • CNNs in brief
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  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
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  • Setting it up
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  • The MNIST dataset again
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  • Strong correlations
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  • Layers of a CNN
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  • Systematic reduction
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  • Importing Keras and Tensorflow
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  • Using TensorFlow backend
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  • Train the model
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  • Visualizing the results
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  • Running with Keras
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  • Final part
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  • Final visualization
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  • Fun links
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  • Applications: solving ordinary differential equations with Neural Networks
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  • Trial solution
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  • More details
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  • Reformulating the problem
    • -
  • Estimating errors
    • -
  • Creating a simple Deep Neural Net
    • -
  • Setting up the code, feed forward part
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  • Backpropagation
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  • Gradient Descent
    • -
  • More on GD and cost function
    • -
  • An implementation of a Deep Neural Network
    • -
  • The final parts of the code
    • -
  • And adding Back propagation
    • -
  • Solving the ODE
    • -
  • Using neural network
    • -
  • Using a deep neural network
    • -
  • Wrapping it up
    • @@ -493,7 +397,7 @@ lmbd = 0.0155
  • 56
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs047.html b/doc/pub/NeuralNet/html/._NeuralNet-bs047.html index a043f7dc3..3f39e3ee9 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs047.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs047.html @@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source - + -Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks +Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning @@ -141,65 +141,7 @@ Automatically generated HTML file from DocOnce source ('Limitations of supervised learning with deep networks', 2, None, - '___sec65'), - ('Convolutional Neural Networks (recognizing images)', - 2, - None, - '___sec66'), - ('Regular NNs don’t scale well to full images', - 2, - None, - '___sec67'), - ('3D volumes of neurons', 2, None, '___sec68'), - ('Layers used to build CNNs', 2, None, '___sec69'), - ('Transforming images', 2, None, '___sec70'), - ('CNNs in brief', 2, None, '___sec71'), - ('CNNs in more detail, building convolutional neural networks in ' - 'Tensorflow and Keras', - 2, - None, - '___sec72'), - ('Setting it up', 2, None, '___sec73'), - ('The MNIST dataset again', 2, None, '___sec74'), - ('Strong correlations', 2, None, '___sec75'), - ('Layers of a CNN', 2, None, '___sec76'), - ('Systematic reduction', 2, None, '___sec77'), - ('Prerequisites: Collect and pre-process data', - 2, - None, - '___sec78'), - ('Importing Keras and Tensorflow', 2, None, '___sec79'), - ('Using TensorFlow backend', 2, None, '___sec80'), - ('Train the model', 2, None, '___sec81'), - ('Visualizing the results', 2, None, '___sec82'), - ('Running with Keras', 2, None, '___sec83'), - ('Final part', 2, None, '___sec84'), - ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86'), - ('Applications: solving ordinary differential equations with ' - 'Neural Networks', - 2, - None, - '___sec87'), - ('Trial solution', 2, None, '___sec88'), - ('More details', 2, None, '___sec89'), - ('Reformulating the problem', 2, None, '___sec90'), - ('Estimating errors', 2, None, '___sec91'), - ('Creating a simple Deep Neural Net', 2, None, '___sec92'), - ('Setting up the code, feed forward part', 2, None, '___sec93'), - ('Backpropagation', 2, None, '___sec94'), - ('Gradient Descent', 2, None, '___sec95'), - ('More on GD and cost function', 2, None, '___sec96'), - ('An implementation of a Deep Neural Network', - 2, - None, - '___sec97'), - ('The final parts of the code', 2, None, '___sec98'), - ('And adding Back propagation', 2, None, '___sec99'), - ('Solving the ODE', 2, None, '___sec100'), - ('Using neural network', 2, None, '___sec101'), - ('Using a deep neural network', 2, None, '___sec102'), - ('Wrapping it up', 2, None, '___sec103')]} + '___sec65')]} end of tocinfo --> @@ -229,7 +171,7 @@ MathJax.Hub.Config({ - Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks + Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning
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  • Which activation function should we use?
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  • Limitations of supervised learning with deep networks
    • -
  • Convolutional Neural Networks (recognizing images)
    • -
  • Regular NNs don’t scale well to full images
    • -
  • 3D volumes of neurons
    • -
  • Layers used to build CNNs
    • -
  • Transforming images
    • -
  • CNNs in brief
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
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  • Setting it up
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  • The MNIST dataset again
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  • Strong correlations
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  • Layers of a CNN
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  • Importing Keras and Tensorflow
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  • Using TensorFlow backend
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  • Visualizing the results
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  • Running with Keras
    • -
  • Final part
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  • Final visualization
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  • Fun links
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  • Applications: solving ordinary differential equations with Neural Networks
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  • Trial solution
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  • More details
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  • Reformulating the problem
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  • Estimating errors
    • -
  • Creating a simple Deep Neural Net
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  • Setting up the code, feed forward part
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  • Backpropagation
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  • Gradient Descent
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  • More on GD and cost function
    • -
  • An implementation of a Deep Neural Network
    • -
  • The final parts of the code
    • -
  • And adding Back propagation
    • -
  • Solving the ODE
    • -
  • Using neural network
    • -
  • Using a deep neural network
    • -
  • Wrapping it up
    • @@ -399,7 +303,7 @@ Andrew Ng goes through some of these considerations in this 56
  • 57
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs048.html b/doc/pub/NeuralNet/html/._NeuralNet-bs048.html index a19219e49..30686ae56 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs048.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs048.html @@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source - + -Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks +Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning @@ -141,65 +141,7 @@ Automatically generated HTML file from DocOnce source ('Limitations of supervised learning with deep networks', 2, None, - '___sec65'), - ('Convolutional Neural Networks (recognizing images)', - 2, - None, - '___sec66'), - ('Regular NNs don’t scale well to full images', - 2, - None, - '___sec67'), - ('3D volumes of neurons', 2, None, '___sec68'), - ('Layers used to build CNNs', 2, None, '___sec69'), - ('Transforming images', 2, None, '___sec70'), - ('CNNs in brief', 2, None, '___sec71'), - ('CNNs in more detail, building convolutional neural networks in ' - 'Tensorflow and Keras', - 2, - None, - '___sec72'), - ('Setting it up', 2, None, '___sec73'), - ('The MNIST dataset again', 2, None, '___sec74'), - ('Strong correlations', 2, None, '___sec75'), - ('Layers of a CNN', 2, None, '___sec76'), - ('Systematic reduction', 2, None, '___sec77'), - ('Prerequisites: Collect and pre-process data', - 2, - None, - '___sec78'), - ('Importing Keras and Tensorflow', 2, None, '___sec79'), - ('Using TensorFlow backend', 2, None, '___sec80'), - ('Train the model', 2, None, '___sec81'), - ('Visualizing the results', 2, None, '___sec82'), - ('Running with Keras', 2, None, '___sec83'), - ('Final part', 2, None, '___sec84'), - ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86'), - ('Applications: solving ordinary differential equations with ' - 'Neural Networks', - 2, - None, - '___sec87'), - ('Trial solution', 2, None, '___sec88'), - ('More details', 2, None, '___sec89'), - ('Reformulating the problem', 2, None, '___sec90'), - ('Estimating errors', 2, None, '___sec91'), - ('Creating a simple Deep Neural Net', 2, None, '___sec92'), - ('Setting up the code, feed forward part', 2, None, '___sec93'), - ('Backpropagation', 2, None, '___sec94'), - ('Gradient Descent', 2, None, '___sec95'), - ('More on GD and cost function', 2, None, '___sec96'), - ('An implementation of a Deep Neural Network', - 2, - None, - '___sec97'), - ('The final parts of the code', 2, None, '___sec98'), - ('And adding Back propagation', 2, None, '___sec99'), - ('Solving the ODE', 2, None, '___sec100'), - ('Using neural network', 2, None, '___sec101'), - ('Using a deep neural network', 2, None, '___sec102'), - ('Wrapping it up', 2, None, '___sec103')]} + '___sec65')]} end of tocinfo --> @@ -229,7 +171,7 @@ MathJax.Hub.Config({ - Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks + Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning
    @@ -303,44 +245,6 @@ MathJax.Hub.Config({
  • Which activation function should we use?
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    • -
  • Convolutional Neural Networks (recognizing images)
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  • Regular NNs don’t scale well to full images
    • -
  • 3D volumes of neurons
    • -
  • Layers used to build CNNs
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  • Transforming images
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  • CNNs in brief
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
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  • Setting it up
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  • The MNIST dataset again
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  • Strong correlations
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  • Layers of a CNN
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  • Systematic reduction
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  • Prerequisites: Collect and pre-process data
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  • Importing Keras and Tensorflow
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  • Using TensorFlow backend
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  • Train the model
    • -
  • Visualizing the results
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  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
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  • Fun links
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  • Applications: solving ordinary differential equations with Neural Networks
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  • Trial solution
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  • More details
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  • Reformulating the problem
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  • Estimating errors
    • -
  • Creating a simple Deep Neural Net
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  • Setting up the code, feed forward part
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  • Backpropagation
    • -
  • Gradient Descent
    • -
  • More on GD and cost function
    • -
  • An implementation of a Deep Neural Network
    • -
  • The final parts of the code
    • -
  • And adding Back propagation
    • -
  • Solving the ODE
    • -
  • Using neural network
    • -
  • Using a deep neural network
    • -
  • Wrapping it up
    • @@ -491,7 +395,7 @@ being realizations of this object with different hyperparameters. An implementat
  • 57
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs049.html b/doc/pub/NeuralNet/html/._NeuralNet-bs049.html index f0dfefaaf..896535792 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs049.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs049.html @@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source - + -Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks +Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning @@ -141,65 +141,7 @@ Automatically generated HTML file from DocOnce source ('Limitations of supervised learning with deep networks', 2, None, - '___sec65'), - ('Convolutional Neural Networks (recognizing images)', - 2, - None, - '___sec66'), - ('Regular NNs don’t scale well to full images', - 2, - None, - '___sec67'), - ('3D volumes of neurons', 2, None, '___sec68'), - ('Layers used to build CNNs', 2, None, '___sec69'), - ('Transforming images', 2, None, '___sec70'), - ('CNNs in brief', 2, None, '___sec71'), - ('CNNs in more detail, building convolutional neural networks in ' - 'Tensorflow and Keras', - 2, - None, - '___sec72'), - ('Setting it up', 2, None, '___sec73'), - ('The MNIST dataset again', 2, None, '___sec74'), - ('Strong correlations', 2, None, '___sec75'), - ('Layers of a CNN', 2, None, '___sec76'), - ('Systematic reduction', 2, None, '___sec77'), - ('Prerequisites: Collect and pre-process data', - 2, - None, - '___sec78'), - ('Importing Keras and Tensorflow', 2, None, '___sec79'), - ('Using TensorFlow backend', 2, None, '___sec80'), - ('Train the model', 2, None, '___sec81'), - ('Visualizing the results', 2, None, '___sec82'), - ('Running with Keras', 2, None, '___sec83'), - ('Final part', 2, None, '___sec84'), - ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86'), - ('Applications: solving ordinary differential equations with ' - 'Neural Networks', - 2, - None, - '___sec87'), - ('Trial solution', 2, None, '___sec88'), - ('More details', 2, None, '___sec89'), - ('Reformulating the problem', 2, None, '___sec90'), - ('Estimating errors', 2, None, '___sec91'), - ('Creating a simple Deep Neural Net', 2, None, '___sec92'), - ('Setting up the code, feed forward part', 2, None, '___sec93'), - ('Backpropagation', 2, None, '___sec94'), - ('Gradient Descent', 2, None, '___sec95'), - ('More on GD and cost function', 2, None, '___sec96'), - ('An implementation of a Deep Neural Network', - 2, - None, - '___sec97'), - ('The final parts of the code', 2, None, '___sec98'), - ('And adding Back propagation', 2, None, '___sec99'), - ('Solving the ODE', 2, None, '___sec100'), - ('Using neural network', 2, None, '___sec101'), - ('Using a deep neural network', 2, None, '___sec102'), - ('Wrapping it up', 2, None, '___sec103')]} + '___sec65')]} end of tocinfo --> @@ -229,7 +171,7 @@ MathJax.Hub.Config({ - Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks + Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning
    @@ -303,44 +245,6 @@ MathJax.Hub.Config({
  • Which activation function should we use?
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  • Limitations of supervised learning with deep networks
    • -
  • Convolutional Neural Networks (recognizing images)
    • -
  • Regular NNs don’t scale well to full images
    • -
  • 3D volumes of neurons
    • -
  • Layers used to build CNNs
    • -
  • Transforming images
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  • CNNs in brief
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
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  • Setting it up
    • -
  • The MNIST dataset again
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  • Strong correlations
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  • Layers of a CNN
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  • Systematic reduction
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  • Prerequisites: Collect and pre-process data
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  • Using TensorFlow backend
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  • Visualizing the results
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  • Running with Keras
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  • Final part
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  • Final visualization
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  • Fun links
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  • Applications: solving ordinary differential equations with Neural Networks
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  • Trial solution
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  • Creating a simple Deep Neural Net
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  • Backpropagation
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  • Gradient Descent
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  • More on GD and cost function
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  • An implementation of a Deep Neural Network
    • -
  • The final parts of the code
    • -
  • And adding Back propagation
    • -
  • Solving the ODE
    • -
  • Using neural network
    • -
  • Using a deep neural network
    • -
  • Wrapping it up
    • @@ -414,7 +318,7 @@ test_predict = dnn58
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs050.html b/doc/pub/NeuralNet/html/._NeuralNet-bs050.html index 8b2ff7147..88deb1c34 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs050.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs050.html @@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source - + -Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks +Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning @@ -141,65 +141,7 @@ Automatically generated HTML file from DocOnce source ('Limitations of supervised learning with deep networks', 2, None, - '___sec65'), - ('Convolutional Neural Networks (recognizing images)', - 2, - None, - '___sec66'), - ('Regular NNs don’t scale well to full images', - 2, - None, - '___sec67'), - ('3D volumes of neurons', 2, None, '___sec68'), - ('Layers used to build CNNs', 2, None, '___sec69'), - ('Transforming images', 2, None, '___sec70'), - ('CNNs in brief', 2, None, '___sec71'), - ('CNNs in more detail, building convolutional neural networks in ' - 'Tensorflow and Keras', - 2, - None, - '___sec72'), - ('Setting it up', 2, None, '___sec73'), - ('The MNIST dataset again', 2, None, '___sec74'), - ('Strong correlations', 2, None, '___sec75'), - ('Layers of a CNN', 2, None, '___sec76'), - ('Systematic reduction', 2, None, '___sec77'), - ('Prerequisites: Collect and pre-process data', - 2, - None, - '___sec78'), - ('Importing Keras and Tensorflow', 2, None, '___sec79'), - ('Using TensorFlow backend', 2, None, '___sec80'), - ('Train the model', 2, None, '___sec81'), - ('Visualizing the results', 2, None, '___sec82'), - ('Running with Keras', 2, None, '___sec83'), - ('Final part', 2, None, '___sec84'), - ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86'), - ('Applications: solving ordinary differential equations with ' - 'Neural Networks', - 2, - None, - '___sec87'), - ('Trial solution', 2, None, '___sec88'), - ('More details', 2, None, '___sec89'), - ('Reformulating the problem', 2, None, '___sec90'), - ('Estimating errors', 2, None, '___sec91'), - ('Creating a simple Deep Neural Net', 2, None, '___sec92'), - ('Setting up the code, feed forward part', 2, None, '___sec93'), - ('Backpropagation', 2, None, '___sec94'), - ('Gradient Descent', 2, None, '___sec95'), - ('More on GD and cost function', 2, None, '___sec96'), - ('An implementation of a Deep Neural Network', - 2, - None, - '___sec97'), - ('The final parts of the code', 2, None, '___sec98'), - ('And adding Back propagation', 2, None, '___sec99'), - ('Solving the ODE', 2, None, '___sec100'), - ('Using neural network', 2, None, '___sec101'), - ('Using a deep neural network', 2, None, '___sec102'), - ('Wrapping it up', 2, None, '___sec103')]} + '___sec65')]} end of tocinfo --> @@ -229,7 +171,7 @@ MathJax.Hub.Config({ - Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks + Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning
    @@ -303,44 +245,6 @@ MathJax.Hub.Config({
  • Which activation function should we use?
  • A top-down perspective on Neural networks
  • Limitations of supervised learning with deep networks
    • -
  • Convolutional Neural Networks (recognizing images)
    • -
  • Regular NNs don’t scale well to full images
    • -
  • 3D volumes of neurons
    • -
  • Layers used to build CNNs
    • -
  • Transforming images
    • -
  • CNNs in brief
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Using TensorFlow backend
    • -
  • Train the model
    • -
  • Visualizing the results
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • Fun links
    • -
  • Applications: solving ordinary differential equations with Neural Networks
    • -
  • Trial solution
    • -
  • More details
    • -
  • Reformulating the problem
    • -
  • Estimating errors
    • -
  • Creating a simple Deep Neural Net
    • -
  • Setting up the code, feed forward part
    • -
  • Backpropagation
    • -
  • Gradient Descent
    • -
  • More on GD and cost function
    • -
  • An implementation of a Deep Neural Network
    • -
  • The final parts of the code
    • -
  • And adding Back propagation
    • -
  • Solving the ODE
    • -
  • Using neural network
    • -
  • Using a deep neural network
    • -
  • Wrapping it up
    • @@ -412,7 +316,7 @@ DNN_numpy = np.
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs051.html b/doc/pub/NeuralNet/html/._NeuralNet-bs051.html index 272cb5712..f83d534ac 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs051.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs051.html @@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source - + -Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks +Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning @@ -141,65 +141,7 @@ Automatically generated HTML file from DocOnce source ('Limitations of supervised learning with deep networks', 2, None, - '___sec65'), - ('Convolutional Neural Networks (recognizing images)', - 2, - None, - '___sec66'), - ('Regular NNs don’t scale well to full images', - 2, - None, - '___sec67'), - ('3D volumes of neurons', 2, None, '___sec68'), - ('Layers used to build CNNs', 2, None, '___sec69'), - ('Transforming images', 2, None, '___sec70'), - ('CNNs in brief', 2, None, '___sec71'), - ('CNNs in more detail, building convolutional neural networks in ' - 'Tensorflow and Keras', - 2, - None, - '___sec72'), - ('Setting it up', 2, None, '___sec73'), - ('The MNIST dataset again', 2, None, '___sec74'), - ('Strong correlations', 2, None, '___sec75'), - ('Layers of a CNN', 2, None, '___sec76'), - ('Systematic reduction', 2, None, '___sec77'), - ('Prerequisites: Collect and pre-process data', - 2, - None, - '___sec78'), - ('Importing Keras and Tensorflow', 2, None, '___sec79'), - ('Using TensorFlow backend', 2, None, '___sec80'), - ('Train the model', 2, None, '___sec81'), - ('Visualizing the results', 2, None, '___sec82'), - ('Running with Keras', 2, None, '___sec83'), - ('Final part', 2, None, '___sec84'), - ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86'), - ('Applications: solving ordinary differential equations with ' - 'Neural Networks', - 2, - None, - '___sec87'), - ('Trial solution', 2, None, '___sec88'), - ('More details', 2, None, '___sec89'), - ('Reformulating the problem', 2, None, '___sec90'), - ('Estimating errors', 2, None, '___sec91'), - ('Creating a simple Deep Neural Net', 2, None, '___sec92'), - ('Setting up the code, feed forward part', 2, None, '___sec93'), - ('Backpropagation', 2, None, '___sec94'), - ('Gradient Descent', 2, None, '___sec95'), - ('More on GD and cost function', 2, None, '___sec96'), - ('An implementation of a Deep Neural Network', - 2, - None, - '___sec97'), - ('The final parts of the code', 2, None, '___sec98'), - ('And adding Back propagation', 2, None, '___sec99'), - ('Solving the ODE', 2, None, '___sec100'), - ('Using neural network', 2, None, '___sec101'), - ('Using a deep neural network', 2, None, '___sec102'), - ('Wrapping it up', 2, None, '___sec103')]} + '___sec65')]} end of tocinfo --> @@ -229,7 +171,7 @@ MathJax.Hub.Config({ - Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks + Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning
    @@ -303,44 +245,6 @@ MathJax.Hub.Config({
  • Which activation function should we use?
  • A top-down perspective on Neural networks
  • Limitations of supervised learning with deep networks
    • -
  • Convolutional Neural Networks (recognizing images)
    • -
  • Regular NNs don’t scale well to full images
    • -
  • 3D volumes of neurons
    • -
  • Layers used to build CNNs
    • -
  • Transforming images
    • -
  • CNNs in brief
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
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  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Using TensorFlow backend
    • -
  • Train the model
    • -
  • Visualizing the results
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • Fun links
    • -
  • Applications: solving ordinary differential equations with Neural Networks
    • -
  • Trial solution
    • -
  • More details
    • -
  • Reformulating the problem
    • -
  • Estimating errors
    • -
  • Creating a simple Deep Neural Net
    • -
  • Setting up the code, feed forward part
    • -
  • Backpropagation
    • -
  • Gradient Descent
    • -
  • More on GD and cost function
    • -
  • An implementation of a Deep Neural Network
    • -
  • The final parts of the code
    • -
  • And adding Back propagation
    • -
  • Solving the ODE
    • -
  • Using neural network
    • -
  • Using a deep neural network
    • -
  • Wrapping it up
    • @@ -421,7 +325,7 @@ plt.show()
  • 60
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs052.html b/doc/pub/NeuralNet/html/._NeuralNet-bs052.html index 84e011c0f..4120cf415 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs052.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs052.html @@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source - + -Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks +Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning @@ -141,65 +141,7 @@ Automatically generated HTML file from DocOnce source ('Limitations of supervised learning with deep networks', 2, None, - '___sec65'), - ('Convolutional Neural Networks (recognizing images)', - 2, - None, - '___sec66'), - ('Regular NNs don’t scale well to full images', - 2, - None, - '___sec67'), - ('3D volumes of neurons', 2, None, '___sec68'), - ('Layers used to build CNNs', 2, None, '___sec69'), - ('Transforming images', 2, None, '___sec70'), - ('CNNs in brief', 2, None, '___sec71'), - ('CNNs in more detail, building convolutional neural networks in ' - 'Tensorflow and Keras', - 2, - None, - '___sec72'), - ('Setting it up', 2, None, '___sec73'), - ('The MNIST dataset again', 2, None, '___sec74'), - ('Strong correlations', 2, None, '___sec75'), - ('Layers of a CNN', 2, None, '___sec76'), - ('Systematic reduction', 2, None, '___sec77'), - ('Prerequisites: Collect and pre-process data', - 2, - None, - '___sec78'), - ('Importing Keras and Tensorflow', 2, None, '___sec79'), - ('Using TensorFlow backend', 2, None, '___sec80'), - ('Train the model', 2, None, '___sec81'), - ('Visualizing the results', 2, None, '___sec82'), - ('Running with Keras', 2, None, '___sec83'), - ('Final part', 2, None, '___sec84'), - ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86'), - ('Applications: solving ordinary differential equations with ' - 'Neural Networks', - 2, - None, - '___sec87'), - ('Trial solution', 2, None, '___sec88'), - ('More details', 2, None, '___sec89'), - ('Reformulating the problem', 2, None, '___sec90'), - ('Estimating errors', 2, None, '___sec91'), - ('Creating a simple Deep Neural Net', 2, None, '___sec92'), - ('Setting up the code, feed forward part', 2, None, '___sec93'), - ('Backpropagation', 2, None, '___sec94'), - ('Gradient Descent', 2, None, '___sec95'), - ('More on GD and cost function', 2, None, '___sec96'), - ('An implementation of a Deep Neural Network', - 2, - None, - '___sec97'), - ('The final parts of the code', 2, None, '___sec98'), - ('And adding Back propagation', 2, None, '___sec99'), - ('Solving the ODE', 2, None, '___sec100'), - ('Using neural network', 2, None, '___sec101'), - ('Using a deep neural network', 2, None, '___sec102'), - ('Wrapping it up', 2, None, '___sec103')]} + '___sec65')]} end of tocinfo --> @@ -229,7 +171,7 @@ MathJax.Hub.Config({ - Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks + Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning
    @@ -303,44 +245,6 @@ MathJax.Hub.Config({
  • Which activation function should we use?
  • A top-down perspective on Neural networks
  • Limitations of supervised learning with deep networks
    • -
  • Convolutional Neural Networks (recognizing images)
    • -
  • Regular NNs don’t scale well to full images
    • -
  • 3D volumes of neurons
    • -
  • Layers used to build CNNs
    • -
  • Transforming images
    • -
  • CNNs in brief
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Using TensorFlow backend
    • -
  • Train the model
    • -
  • Visualizing the results
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • Fun links
    • -
  • Applications: solving ordinary differential equations with Neural Networks
    • -
  • Trial solution
    • -
  • More details
    • -
  • Reformulating the problem
    • -
  • Estimating errors
    • -
  • Creating a simple Deep Neural Net
    • -
  • Setting up the code, feed forward part
    • -
  • Backpropagation
    • -
  • Gradient Descent
    • -
  • More on GD and cost function
    • -
  • An implementation of a Deep Neural Network
    • -
  • The final parts of the code
    • -
  • And adding Back propagation
    • -
  • Solving the ODE
    • -
  • Using neural network
    • -
  • Using a deep neural network
    • -
  • Wrapping it up
    • @@ -418,7 +322,7 @@ DNN_scikit = np
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs053.html b/doc/pub/NeuralNet/html/._NeuralNet-bs053.html index f5562fb89..34c0ed7d4 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs053.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs053.html @@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source - + -Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks +Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning @@ -141,65 +141,7 @@ Automatically generated HTML file from DocOnce source ('Limitations of supervised learning with deep networks', 2, None, - '___sec65'), - ('Convolutional Neural Networks (recognizing images)', - 2, - None, - '___sec66'), - ('Regular NNs don’t scale well to full images', - 2, - None, - '___sec67'), - ('3D volumes of neurons', 2, None, '___sec68'), - ('Layers used to build CNNs', 2, None, '___sec69'), - ('Transforming images', 2, None, '___sec70'), - ('CNNs in brief', 2, None, '___sec71'), - ('CNNs in more detail, building convolutional neural networks in ' - 'Tensorflow and Keras', - 2, - None, - '___sec72'), - ('Setting it up', 2, None, '___sec73'), - ('The MNIST dataset again', 2, None, '___sec74'), - ('Strong correlations', 2, None, '___sec75'), - ('Layers of a CNN', 2, None, '___sec76'), - ('Systematic reduction', 2, None, '___sec77'), - ('Prerequisites: Collect and pre-process data', - 2, - None, - '___sec78'), - ('Importing Keras and Tensorflow', 2, None, '___sec79'), - ('Using TensorFlow backend', 2, None, '___sec80'), - ('Train the model', 2, None, '___sec81'), - ('Visualizing the results', 2, None, '___sec82'), - ('Running with Keras', 2, None, '___sec83'), - ('Final part', 2, None, '___sec84'), - ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86'), - ('Applications: solving ordinary differential equations with ' - 'Neural Networks', - 2, - None, - '___sec87'), - ('Trial solution', 2, None, '___sec88'), - ('More details', 2, None, '___sec89'), - ('Reformulating the problem', 2, None, '___sec90'), - ('Estimating errors', 2, None, '___sec91'), - ('Creating a simple Deep Neural Net', 2, None, '___sec92'), - ('Setting up the code, feed forward part', 2, None, '___sec93'), - ('Backpropagation', 2, None, '___sec94'), - ('Gradient Descent', 2, None, '___sec95'), - ('More on GD and cost function', 2, None, '___sec96'), - ('An implementation of a Deep Neural Network', - 2, - None, - '___sec97'), - ('The final parts of the code', 2, None, '___sec98'), - ('And adding Back propagation', 2, None, '___sec99'), - ('Solving the ODE', 2, None, '___sec100'), - ('Using neural network', 2, None, '___sec101'), - ('Using a deep neural network', 2, None, '___sec102'), - ('Wrapping it up', 2, None, '___sec103')]} + '___sec65')]} end of tocinfo --> @@ -229,7 +171,7 @@ MathJax.Hub.Config({ - Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks + Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning
    @@ -303,44 +245,6 @@ MathJax.Hub.Config({
  • Which activation function should we use?
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  • Limitations of supervised learning with deep networks
    • -
  • Convolutional Neural Networks (recognizing images)
    • -
  • Regular NNs don’t scale well to full images
    • -
  • 3D volumes of neurons
    • -
  • Layers used to build CNNs
    • -
  • Transforming images
    • -
  • CNNs in brief
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Using TensorFlow backend
    • -
  • Train the model
    • -
  • Visualizing the results
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • Fun links
    • -
  • Applications: solving ordinary differential equations with Neural Networks
    • -
  • Trial solution
    • -
  • More details
    • -
  • Reformulating the problem
    • -
  • Estimating errors
    • -
  • Creating a simple Deep Neural Net
    • -
  • Setting up the code, feed forward part
    • -
  • Backpropagation
    • -
  • Gradient Descent
    • -
  • More on GD and cost function
    • -
  • An implementation of a Deep Neural Network
    • -
  • The final parts of the code
    • -
  • And adding Back propagation
    • -
  • Solving the ODE
    • -
  • Using neural network
    • -
  • Using a deep neural network
    • -
  • Wrapping it up
    • @@ -421,7 +325,7 @@ plt.show()
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs054.html b/doc/pub/NeuralNet/html/._NeuralNet-bs054.html index e8664cde3..7f26b666f 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs054.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs054.html @@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source - + -Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks +Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning @@ -141,65 +141,7 @@ Automatically generated HTML file from DocOnce source ('Limitations of supervised learning with deep networks', 2, None, - '___sec65'), - ('Convolutional Neural Networks (recognizing images)', - 2, - None, - '___sec66'), - ('Regular NNs don’t scale well to full images', - 2, - None, - '___sec67'), - ('3D volumes of neurons', 2, None, '___sec68'), - ('Layers used to build CNNs', 2, None, '___sec69'), - ('Transforming images', 2, None, '___sec70'), - ('CNNs in brief', 2, None, '___sec71'), - ('CNNs in more detail, building convolutional neural networks in ' - 'Tensorflow and Keras', - 2, - None, - '___sec72'), - ('Setting it up', 2, None, '___sec73'), - ('The MNIST dataset again', 2, None, '___sec74'), - ('Strong correlations', 2, None, '___sec75'), - ('Layers of a CNN', 2, None, '___sec76'), - ('Systematic reduction', 2, None, '___sec77'), - ('Prerequisites: Collect and pre-process data', - 2, - None, - '___sec78'), - ('Importing Keras and Tensorflow', 2, None, '___sec79'), - ('Using TensorFlow backend', 2, None, '___sec80'), - ('Train the model', 2, None, '___sec81'), - ('Visualizing the results', 2, None, '___sec82'), - ('Running with Keras', 2, None, '___sec83'), - ('Final part', 2, None, '___sec84'), - ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86'), - ('Applications: solving ordinary differential equations with ' - 'Neural Networks', - 2, - None, - '___sec87'), - ('Trial solution', 2, None, '___sec88'), - ('More details', 2, None, '___sec89'), - ('Reformulating the problem', 2, None, '___sec90'), - ('Estimating errors', 2, None, '___sec91'), - ('Creating a simple Deep Neural Net', 2, None, '___sec92'), - ('Setting up the code, feed forward part', 2, None, '___sec93'), - ('Backpropagation', 2, None, '___sec94'), - ('Gradient Descent', 2, None, '___sec95'), - ('More on GD and cost function', 2, None, '___sec96'), - ('An implementation of a Deep Neural Network', - 2, - None, - '___sec97'), - ('The final parts of the code', 2, None, '___sec98'), - ('And adding Back propagation', 2, None, '___sec99'), - ('Solving the ODE', 2, None, '___sec100'), - ('Using neural network', 2, None, '___sec101'), - ('Using a deep neural network', 2, None, '___sec102'), - ('Wrapping it up', 2, None, '___sec103')]} + '___sec65')]} end of tocinfo --> @@ -229,7 +171,7 @@ MathJax.Hub.Config({ - Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks + Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning
    @@ -303,44 +245,6 @@ MathJax.Hub.Config({
  • Which activation function should we use?
  • A top-down perspective on Neural networks
  • Limitations of supervised learning with deep networks
    • -
  • Convolutional Neural Networks (recognizing images)
    • -
  • Regular NNs don’t scale well to full images
    • -
  • 3D volumes of neurons
    • -
  • Layers used to build CNNs
    • -
  • Transforming images
    • -
  • CNNs in brief
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
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  • Prerequisites: Collect and pre-process data
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  • Importing Keras and Tensorflow
    • -
  • Using TensorFlow backend
    • -
  • Train the model
    • -
  • Visualizing the results
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • Fun links
    • -
  • Applications: solving ordinary differential equations with Neural Networks
    • -
  • Trial solution
    • -
  • More details
    • -
  • Reformulating the problem
    • -
  • Estimating errors
    • -
  • Creating a simple Deep Neural Net
    • -
  • Setting up the code, feed forward part
    • -
  • Backpropagation
    • -
  • Gradient Descent
    • -
  • More on GD and cost function
    • -
  • An implementation of a Deep Neural Network
    • -
  • The final parts of the code
    • -
  • And adding Back propagation
    • -
  • Solving the ODE
    • -
  • Using neural network
    • -
  • Using a deep neural network
    • -
  • Wrapping it up
    • @@ -394,7 +298,7 @@ NumPy arrays.
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs055.html b/doc/pub/NeuralNet/html/._NeuralNet-bs055.html index 72d1a9649..371e2b814 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs055.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs055.html @@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source - + -Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks +Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning @@ -141,65 +141,7 @@ Automatically generated HTML file from DocOnce source ('Limitations of supervised learning with deep networks', 2, None, - '___sec65'), - ('Convolutional Neural Networks (recognizing images)', - 2, - None, - '___sec66'), - ('Regular NNs don’t scale well to full images', - 2, - None, - '___sec67'), - ('3D volumes of neurons', 2, None, '___sec68'), - ('Layers used to build CNNs', 2, None, '___sec69'), - ('Transforming images', 2, None, '___sec70'), - ('CNNs in brief', 2, None, '___sec71'), - ('CNNs in more detail, building convolutional neural networks in ' - 'Tensorflow and Keras', - 2, - None, - '___sec72'), - ('Setting it up', 2, None, '___sec73'), - ('The MNIST dataset again', 2, None, '___sec74'), - ('Strong correlations', 2, None, '___sec75'), - ('Layers of a CNN', 2, None, '___sec76'), - ('Systematic reduction', 2, None, '___sec77'), - ('Prerequisites: Collect and pre-process data', - 2, - None, - '___sec78'), - ('Importing Keras and Tensorflow', 2, None, '___sec79'), - ('Using TensorFlow backend', 2, None, '___sec80'), - ('Train the model', 2, None, '___sec81'), - ('Visualizing the results', 2, None, '___sec82'), - ('Running with Keras', 2, None, '___sec83'), - ('Final part', 2, None, '___sec84'), - ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86'), - ('Applications: solving ordinary differential equations with ' - 'Neural Networks', - 2, - None, - '___sec87'), - ('Trial solution', 2, None, '___sec88'), - ('More details', 2, None, '___sec89'), - ('Reformulating the problem', 2, None, '___sec90'), - ('Estimating errors', 2, None, '___sec91'), - ('Creating a simple Deep Neural Net', 2, None, '___sec92'), - ('Setting up the code, feed forward part', 2, None, '___sec93'), - ('Backpropagation', 2, None, '___sec94'), - ('Gradient Descent', 2, None, '___sec95'), - ('More on GD and cost function', 2, None, '___sec96'), - ('An implementation of a Deep Neural Network', - 2, - None, - '___sec97'), - ('The final parts of the code', 2, None, '___sec98'), - ('And adding Back propagation', 2, None, '___sec99'), - ('Solving the ODE', 2, None, '___sec100'), - ('Using neural network', 2, None, '___sec101'), - ('Using a deep neural network', 2, None, '___sec102'), - ('Wrapping it up', 2, None, '___sec103')]} + '___sec65')]} end of tocinfo --> @@ -229,7 +171,7 @@ MathJax.Hub.Config({ - Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks + Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning
    @@ -303,44 +245,6 @@ MathJax.Hub.Config({
  • Which activation function should we use?
  • A top-down perspective on Neural networks
  • Limitations of supervised learning with deep networks
    • -
  • Convolutional Neural Networks (recognizing images)
    • -
  • Regular NNs don’t scale well to full images
    • -
  • 3D volumes of neurons
    • -
  • Layers used to build CNNs
    • -
  • Transforming images
    • -
  • CNNs in brief
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Using TensorFlow backend
    • -
  • Train the model
    • -
  • Visualizing the results
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • Fun links
    • -
  • Applications: solving ordinary differential equations with Neural Networks
    • -
  • Trial solution
    • -
  • More details
    • -
  • Reformulating the problem
    • -
  • Estimating errors
    • -
  • Creating a simple Deep Neural Net
    • -
  • Setting up the code, feed forward part
    • -
  • Backpropagation
    • -
  • Gradient Descent
    • -
  • More on GD and cost function
    • -
  • An implementation of a Deep Neural Network
    • -
  • The final parts of the code
    • -
  • And adding Back propagation
    • -
  • Solving the ODE
    • -
  • Using neural network
    • -
  • Using a deep neural network
    • -
  • Wrapping it up
    • @@ -424,7 +328,7 @@ and/or if you use anaconda, just write (or install from the graphical use
  • 64
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs056.html b/doc/pub/NeuralNet/html/._NeuralNet-bs056.html index ce5339572..b7e8d3d8e 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs056.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs056.html @@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source - + -Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks +Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning @@ -141,65 +141,7 @@ Automatically generated HTML file from DocOnce source ('Limitations of supervised learning with deep networks', 2, None, - '___sec65'), - ('Convolutional Neural Networks (recognizing images)', - 2, - None, - '___sec66'), - ('Regular NNs don’t scale well to full images', - 2, - None, - '___sec67'), - ('3D volumes of neurons', 2, None, '___sec68'), - ('Layers used to build CNNs', 2, None, '___sec69'), - ('Transforming images', 2, None, '___sec70'), - ('CNNs in brief', 2, None, '___sec71'), - ('CNNs in more detail, building convolutional neural networks in ' - 'Tensorflow and Keras', - 2, - None, - '___sec72'), - ('Setting it up', 2, None, '___sec73'), - ('The MNIST dataset again', 2, None, '___sec74'), - ('Strong correlations', 2, None, '___sec75'), - ('Layers of a CNN', 2, None, '___sec76'), - ('Systematic reduction', 2, None, '___sec77'), - ('Prerequisites: Collect and pre-process data', - 2, - None, - '___sec78'), - ('Importing Keras and Tensorflow', 2, None, '___sec79'), - ('Using TensorFlow backend', 2, None, '___sec80'), - ('Train the model', 2, None, '___sec81'), - ('Visualizing the results', 2, None, '___sec82'), - ('Running with Keras', 2, None, '___sec83'), - ('Final part', 2, None, '___sec84'), - ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86'), - ('Applications: solving ordinary differential equations with ' - 'Neural Networks', - 2, - None, - '___sec87'), - ('Trial solution', 2, None, '___sec88'), - ('More details', 2, None, '___sec89'), - ('Reformulating the problem', 2, None, '___sec90'), - ('Estimating errors', 2, None, '___sec91'), - ('Creating a simple Deep Neural Net', 2, None, '___sec92'), - ('Setting up the code, feed forward part', 2, None, '___sec93'), - ('Backpropagation', 2, None, '___sec94'), - ('Gradient Descent', 2, None, '___sec95'), - ('More on GD and cost function', 2, None, '___sec96'), - ('An implementation of a Deep Neural Network', - 2, - None, - '___sec97'), - ('The final parts of the code', 2, None, '___sec98'), - ('And adding Back propagation', 2, None, '___sec99'), - ('Solving the ODE', 2, None, '___sec100'), - ('Using neural network', 2, None, '___sec101'), - ('Using a deep neural network', 2, None, '___sec102'), - ('Wrapping it up', 2, None, '___sec103')]} + '___sec65')]} end of tocinfo --> @@ -229,7 +171,7 @@ MathJax.Hub.Config({ - Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks + Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning
    @@ -303,44 +245,6 @@ MathJax.Hub.Config({
  • Which activation function should we use?
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  • Limitations of supervised learning with deep networks
    • -
  • Convolutional Neural Networks (recognizing images)
    • -
  • Regular NNs don’t scale well to full images
    • -
  • 3D volumes of neurons
    • -
  • Layers used to build CNNs
    • -
  • Transforming images
    • -
  • CNNs in brief
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
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  • Setting it up
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  • The MNIST dataset again
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  • Strong correlations
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  • Layers of a CNN
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  • Systematic reduction
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  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Using TensorFlow backend
    • -
  • Train the model
    • -
  • Visualizing the results
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • Fun links
    • -
  • Applications: solving ordinary differential equations with Neural Networks
    • -
  • Trial solution
    • -
  • More details
    • -
  • Reformulating the problem
    • -
  • Estimating errors
    • -
  • Creating a simple Deep Neural Net
    • -
  • Setting up the code, feed forward part
    • -
  • Backpropagation
    • -
  • Gradient Descent
    • -
  • More on GD and cost function
    • -
  • An implementation of a Deep Neural Network
    • -
  • The final parts of the code
    • -
  • And adding Back propagation
    • -
  • Solving the ODE
    • -
  • Using neural network
    • -
  • Using a deep neural network
    • -
  • Wrapping it up
    • @@ -445,7 +349,7 @@ X_train, X_test, Y_train, Y_test = train_tes
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs057.html b/doc/pub/NeuralNet/html/._NeuralNet-bs057.html index b7804cbe8..11cdcec8d 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs057.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs057.html @@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source - + -Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks +Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning @@ -141,65 +141,7 @@ Automatically generated HTML file from DocOnce source ('Limitations of supervised learning with deep networks', 2, None, - '___sec65'), - ('Convolutional Neural Networks (recognizing images)', - 2, - None, - '___sec66'), - ('Regular NNs don’t scale well to full images', - 2, - None, - '___sec67'), - ('3D volumes of neurons', 2, None, '___sec68'), - ('Layers used to build CNNs', 2, None, '___sec69'), - ('Transforming images', 2, None, '___sec70'), - ('CNNs in brief', 2, None, '___sec71'), - ('CNNs in more detail, building convolutional neural networks in ' - 'Tensorflow and Keras', - 2, - None, - '___sec72'), - ('Setting it up', 2, None, '___sec73'), - ('The MNIST dataset again', 2, None, '___sec74'), - ('Strong correlations', 2, None, '___sec75'), - ('Layers of a CNN', 2, None, '___sec76'), - ('Systematic reduction', 2, None, '___sec77'), - ('Prerequisites: Collect and pre-process data', - 2, - None, - '___sec78'), - ('Importing Keras and Tensorflow', 2, None, '___sec79'), - ('Using TensorFlow backend', 2, None, '___sec80'), - ('Train the model', 2, None, '___sec81'), - ('Visualizing the results', 2, None, '___sec82'), - ('Running with Keras', 2, None, '___sec83'), - ('Final part', 2, None, '___sec84'), - ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86'), - ('Applications: solving ordinary differential equations with ' - 'Neural Networks', - 2, - None, - '___sec87'), - ('Trial solution', 2, None, '___sec88'), - ('More details', 2, None, '___sec89'), - ('Reformulating the problem', 2, None, '___sec90'), - ('Estimating errors', 2, None, '___sec91'), - ('Creating a simple Deep Neural Net', 2, None, '___sec92'), - ('Setting up the code, feed forward part', 2, None, '___sec93'), - ('Backpropagation', 2, None, '___sec94'), - ('Gradient Descent', 2, None, '___sec95'), - ('More on GD and cost function', 2, None, '___sec96'), - ('An implementation of a Deep Neural Network', - 2, - None, - '___sec97'), - ('The final parts of the code', 2, None, '___sec98'), - ('And adding Back propagation', 2, None, '___sec99'), - ('Solving the ODE', 2, None, '___sec100'), - ('Using neural network', 2, None, '___sec101'), - ('Using a deep neural network', 2, None, '___sec102'), - ('Wrapping it up', 2, None, '___sec103')]} + '___sec65')]} end of tocinfo --> @@ -229,7 +171,7 @@ MathJax.Hub.Config({ - Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks + Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning
    @@ -303,44 +245,6 @@ MathJax.Hub.Config({
  • Which activation function should we use?
  • A top-down perspective on Neural networks
  • Limitations of supervised learning with deep networks
    • -
  • Convolutional Neural Networks (recognizing images)
    • -
  • Regular NNs don’t scale well to full images
    • -
  • 3D volumes of neurons
    • -
  • Layers used to build CNNs
    • -
  • Transforming images
    • -
  • CNNs in brief
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Using TensorFlow backend
    • -
  • Train the model
    • -
  • Visualizing the results
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • Fun links
    • -
  • Applications: solving ordinary differential equations with Neural Networks
    • -
  • Trial solution
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  • More details
    • -
  • Reformulating the problem
    • -
  • Estimating errors
    • -
  • Creating a simple Deep Neural Net
    • -
  • Setting up the code, feed forward part
    • -
  • Backpropagation
    • -
  • Gradient Descent
    • -
  • More on GD and cost function
    • -
  • An implementation of a Deep Neural Network
    • -
  • The final parts of the code
    • -
  • And adding Back propagation
    • -
  • Solving the ODE
    • -
  • Using neural network
    • -
  • Using a deep neural network
    • -
  • Wrapping it up
    • @@ -522,8 +426,6 @@ MathJax.Hub.Config({
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs058.html b/doc/pub/NeuralNet/html/._NeuralNet-bs058.html index 0843f1bdc..0a3e9c5f2 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs058.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs058.html @@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source - + -Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks +Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning @@ -141,65 +141,7 @@ Automatically generated HTML file from DocOnce source ('Limitations of supervised learning with deep networks', 2, None, - '___sec65'), - ('Convolutional Neural Networks (recognizing images)', - 2, - None, - '___sec66'), - ('Regular NNs don’t scale well to full images', - 2, - None, - '___sec67'), - ('3D volumes of neurons', 2, None, '___sec68'), - ('Layers used to build CNNs', 2, None, '___sec69'), - ('Transforming images', 2, None, '___sec70'), - ('CNNs in brief', 2, None, '___sec71'), - ('CNNs in more detail, building convolutional neural networks in ' - 'Tensorflow and Keras', - 2, - None, - '___sec72'), - ('Setting it up', 2, None, '___sec73'), - ('The MNIST dataset again', 2, None, '___sec74'), - ('Strong correlations', 2, None, '___sec75'), - ('Layers of a CNN', 2, None, '___sec76'), - ('Systematic reduction', 2, None, '___sec77'), - ('Prerequisites: Collect and pre-process data', - 2, - None, - '___sec78'), - ('Importing Keras and Tensorflow', 2, None, '___sec79'), - ('Using TensorFlow backend', 2, None, '___sec80'), - ('Train the model', 2, None, '___sec81'), - ('Visualizing the results', 2, None, '___sec82'), - ('Running with Keras', 2, None, '___sec83'), - ('Final part', 2, None, '___sec84'), - ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86'), - ('Applications: solving ordinary differential equations with ' - 'Neural Networks', - 2, - None, - '___sec87'), - ('Trial solution', 2, None, '___sec88'), - ('More details', 2, None, '___sec89'), - ('Reformulating the problem', 2, None, '___sec90'), - ('Estimating errors', 2, None, '___sec91'), - ('Creating a simple Deep Neural Net', 2, None, '___sec92'), - ('Setting up the code, feed forward part', 2, None, '___sec93'), - ('Backpropagation', 2, None, '___sec94'), - ('Gradient Descent', 2, None, '___sec95'), - ('More on GD and cost function', 2, None, '___sec96'), - ('An implementation of a Deep Neural Network', - 2, - None, - '___sec97'), - ('The final parts of the code', 2, None, '___sec98'), - ('And adding Back propagation', 2, None, '___sec99'), - ('Solving the ODE', 2, None, '___sec100'), - ('Using neural network', 2, None, '___sec101'), - ('Using a deep neural network', 2, None, '___sec102'), - ('Wrapping it up', 2, None, '___sec103')]} + '___sec65')]} end of tocinfo --> @@ -229,7 +171,7 @@ MathJax.Hub.Config({ - Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks + Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning
    @@ -303,44 +245,6 @@ MathJax.Hub.Config({
  • Which activation function should we use?
  • A top-down perspective on Neural networks
  • Limitations of supervised learning with deep networks
    • -
  • Convolutional Neural Networks (recognizing images)
    • -
  • Regular NNs don’t scale well to full images
    • -
  • 3D volumes of neurons
    • -
  • Layers used to build CNNs
    • -
  • Transforming images
    • -
  • CNNs in brief
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Using TensorFlow backend
    • -
  • Train the model
    • -
  • Visualizing the results
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • Fun links
    • -
  • Applications: solving ordinary differential equations with Neural Networks
    • -
  • Trial solution
    • -
  • More details
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  • Reformulating the problem
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  • Estimating errors
    • -
  • Creating a simple Deep Neural Net
    • -
  • Setting up the code, feed forward part
    • -
  • Backpropagation
    • -
  • Gradient Descent
    • -
  • More on GD and cost function
    • -
  • An implementation of a Deep Neural Network
    • -
  • The final parts of the code
    • -
  • And adding Back propagation
    • -
  • Solving the ODE
    • -
  • Using neural network
    • -
  • Using a deep neural network
    • -
  • Wrapping it up
    • @@ -455,9 +359,6 @@ writer.add_graph(tf65
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs059.html b/doc/pub/NeuralNet/html/._NeuralNet-bs059.html index 0fb319d24..6675d248a 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs059.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs059.html @@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source - + -Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks +Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning @@ -141,65 +141,7 @@ Automatically generated HTML file from DocOnce source ('Limitations of supervised learning with deep networks', 2, None, - '___sec65'), - ('Convolutional Neural Networks (recognizing images)', - 2, - None, - '___sec66'), - ('Regular NNs don’t scale well to full images', - 2, - None, - '___sec67'), - ('3D volumes of neurons', 2, None, '___sec68'), - ('Layers used to build CNNs', 2, None, '___sec69'), - ('Transforming images', 2, None, '___sec70'), - ('CNNs in brief', 2, None, '___sec71'), - ('CNNs in more detail, building convolutional neural networks in ' - 'Tensorflow and Keras', - 2, - None, - '___sec72'), - ('Setting it up', 2, None, '___sec73'), - ('The MNIST dataset again', 2, None, '___sec74'), - ('Strong correlations', 2, None, '___sec75'), - ('Layers of a CNN', 2, None, '___sec76'), - ('Systematic reduction', 2, None, '___sec77'), - ('Prerequisites: Collect and pre-process data', - 2, - None, - '___sec78'), - ('Importing Keras and Tensorflow', 2, None, '___sec79'), - ('Using TensorFlow backend', 2, None, '___sec80'), - ('Train the model', 2, None, '___sec81'), - ('Visualizing the results', 2, None, '___sec82'), - ('Running with Keras', 2, None, '___sec83'), - ('Final part', 2, None, '___sec84'), - ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86'), - ('Applications: solving ordinary differential equations with ' - 'Neural Networks', - 2, - None, - '___sec87'), - ('Trial solution', 2, None, '___sec88'), - ('More details', 2, None, '___sec89'), - ('Reformulating the problem', 2, None, '___sec90'), - ('Estimating errors', 2, None, '___sec91'), - ('Creating a simple Deep Neural Net', 2, None, '___sec92'), - ('Setting up the code, feed forward part', 2, None, '___sec93'), - ('Backpropagation', 2, None, '___sec94'), - ('Gradient Descent', 2, None, '___sec95'), - ('More on GD and cost function', 2, None, '___sec96'), - ('An implementation of a Deep Neural Network', - 2, - None, - '___sec97'), - ('The final parts of the code', 2, None, '___sec98'), - ('And adding Back propagation', 2, None, '___sec99'), - ('Solving the ODE', 2, None, '___sec100'), - ('Using neural network', 2, None, '___sec101'), - ('Using a deep neural network', 2, None, '___sec102'), - ('Wrapping it up', 2, None, '___sec103')]} + '___sec65')]} end of tocinfo --> @@ -229,7 +171,7 @@ MathJax.Hub.Config({ - Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks + Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning
    @@ -303,44 +245,6 @@ MathJax.Hub.Config({
  • Which activation function should we use?
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  • Limitations of supervised learning with deep networks
    • -
  • Convolutional Neural Networks (recognizing images)
    • -
  • Regular NNs don’t scale well to full images
    • -
  • 3D volumes of neurons
    • -
  • Layers used to build CNNs
    • -
  • Transforming images
    • -
  • CNNs in brief
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
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  • Prerequisites: Collect and pre-process data
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  • Importing Keras and Tensorflow
    • -
  • Using TensorFlow backend
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  • Train the model
    • -
  • Visualizing the results
    • -
  • Running with Keras
    • -
  • Final part
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  • Final visualization
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  • Fun links
    • -
  • Applications: solving ordinary differential equations with Neural Networks
    • -
  • Trial solution
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  • More details
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  • Creating a simple Deep Neural Net
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  • Setting up the code, feed forward part
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  • Backpropagation
    • -
  • Gradient Descent
    • -
  • More on GD and cost function
    • -
  • An implementation of a Deep Neural Network
    • -
  • The final parts of the code
    • -
  • And adding Back propagation
    • -
  • Solving the ODE
    • -
  • Using neural network
    • -
  • Using a deep neural network
    • -
  • Wrapping it up
    • @@ -475,10 +379,6 @@ plt.show()
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  • The final parts of the code
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  • Solving the ODE
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    • @@ -404,11 +308,6 @@ learn at widely different speeds
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  • Using a deep neural network
    • -
  • Wrapping it up
    • @@ -391,15 +295,6 @@ bootstrap to evaluate other activation functions.
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs065.html b/doc/pub/NeuralNet/html/._NeuralNet-bs065.html index c92db07b2..334e9b551 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs065.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs065.html @@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source - + -Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks +Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning @@ -141,65 +141,7 @@ Automatically generated HTML file from DocOnce source ('Limitations of supervised learning with deep networks', 2, None, - '___sec65'), - ('Convolutional Neural Networks (recognizing images)', - 2, - None, - '___sec66'), - ('Regular NNs don’t scale well to full images', - 2, - None, - '___sec67'), - ('3D volumes of neurons', 2, None, '___sec68'), - ('Layers used to build CNNs', 2, None, '___sec69'), - ('Transforming images', 2, None, '___sec70'), - ('CNNs in brief', 2, None, '___sec71'), - ('CNNs in more detail, building convolutional neural networks in ' - 'Tensorflow and Keras', - 2, - None, - '___sec72'), - ('Setting it up', 2, None, '___sec73'), - ('The MNIST dataset again', 2, None, '___sec74'), - ('Strong correlations', 2, None, '___sec75'), - ('Layers of a CNN', 2, None, '___sec76'), - ('Systematic reduction', 2, None, '___sec77'), - ('Prerequisites: Collect and pre-process data', - 2, - None, - '___sec78'), - ('Importing Keras and Tensorflow', 2, None, '___sec79'), - ('Using TensorFlow backend', 2, None, '___sec80'), - ('Train the model', 2, None, '___sec81'), - ('Visualizing the results', 2, None, '___sec82'), - ('Running with Keras', 2, None, '___sec83'), - ('Final part', 2, None, '___sec84'), - ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86'), - ('Applications: solving ordinary differential equations with ' - 'Neural Networks', - 2, - None, - '___sec87'), - ('Trial solution', 2, None, '___sec88'), - ('More details', 2, None, '___sec89'), - ('Reformulating the problem', 2, None, '___sec90'), - ('Estimating errors', 2, None, '___sec91'), - ('Creating a simple Deep Neural Net', 2, None, '___sec92'), - ('Setting up the code, feed forward part', 2, None, '___sec93'), - ('Backpropagation', 2, None, '___sec94'), - ('Gradient Descent', 2, None, '___sec95'), - ('More on GD and cost function', 2, None, '___sec96'), - ('An implementation of a Deep Neural Network', - 2, - None, - '___sec97'), - ('The final parts of the code', 2, None, '___sec98'), - ('And adding Back propagation', 2, None, '___sec99'), - ('Solving the ODE', 2, None, '___sec100'), - ('Using neural network', 2, None, '___sec101'), - ('Using a deep neural network', 2, None, '___sec102'), - ('Wrapping it up', 2, None, '___sec103')]} + '___sec65')]} end of tocinfo --> @@ -229,7 +171,7 @@ MathJax.Hub.Config({ - Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks + Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning
    @@ -303,44 +245,6 @@ MathJax.Hub.Config({
  • Which activation function should we use?
  • A top-down perspective on Neural networks
  • Limitations of supervised learning with deep networks
    • -
  • Convolutional Neural Networks (recognizing images)
    • -
  • Regular NNs don’t scale well to full images
    • -
  • 3D volumes of neurons
    • -
  • Layers used to build CNNs
    • -
  • Transforming images
    • -
  • CNNs in brief
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Using TensorFlow backend
    • -
  • Train the model
    • -
  • Visualizing the results
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • Fun links
    • -
  • Applications: solving ordinary differential equations with Neural Networks
    • -
  • Trial solution
    • -
  • More details
    • -
  • Reformulating the problem
    • -
  • Estimating errors
    • -
  • Creating a simple Deep Neural Net
    • -
  • Setting up the code, feed forward part
    • -
  • Backpropagation
    • -
  • Gradient Descent
    • -
  • More on GD and cost function
    • -
  • An implementation of a Deep Neural Network
    • -
  • The final parts of the code
    • -
  • And adding Back propagation
    • -
  • Solving the ODE
    • -
  • Using neural network
    • -
  • Using a deep neural network
    • -
  • Wrapping it up
    • @@ -412,16 +316,6 @@ supervised learning.
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs066.html b/doc/pub/NeuralNet/html/._NeuralNet-bs066.html index 26e51620a..41f5912b9 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs066.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs066.html @@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source - + -Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks +Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning @@ -141,65 +141,7 @@ Automatically generated HTML file from DocOnce source ('Limitations of supervised learning with deep networks', 2, None, - '___sec65'), - ('Convolutional Neural Networks (recognizing images)', - 2, - None, - '___sec66'), - ('Regular NNs don’t scale well to full images', - 2, - None, - '___sec67'), - ('3D volumes of neurons', 2, None, '___sec68'), - ('Layers used to build CNNs', 2, None, '___sec69'), - ('Transforming images', 2, None, '___sec70'), - ('CNNs in brief', 2, None, '___sec71'), - ('CNNs in more detail, building convolutional neural networks in ' - 'Tensorflow and Keras', - 2, - None, - '___sec72'), - ('Setting it up', 2, None, '___sec73'), - ('The MNIST dataset again', 2, None, '___sec74'), - ('Strong correlations', 2, None, '___sec75'), - ('Layers of a CNN', 2, None, '___sec76'), - ('Systematic reduction', 2, None, '___sec77'), - ('Prerequisites: Collect and pre-process data', - 2, - None, - '___sec78'), - ('Importing Keras and Tensorflow', 2, None, '___sec79'), - ('Using TensorFlow backend', 2, None, '___sec80'), - ('Train the model', 2, None, '___sec81'), - ('Visualizing the results', 2, None, '___sec82'), - ('Running with Keras', 2, None, '___sec83'), - ('Final part', 2, None, '___sec84'), - ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86'), - ('Applications: solving ordinary differential equations with ' - 'Neural Networks', - 2, - None, - '___sec87'), - ('Trial solution', 2, None, '___sec88'), - ('More details', 2, None, '___sec89'), - ('Reformulating the problem', 2, None, '___sec90'), - ('Estimating errors', 2, None, '___sec91'), - ('Creating a simple Deep Neural Net', 2, None, '___sec92'), - ('Setting up the code, feed forward part', 2, None, '___sec93'), - ('Backpropagation', 2, None, '___sec94'), - ('Gradient Descent', 2, None, '___sec95'), - ('More on GD and cost function', 2, None, '___sec96'), - ('An implementation of a Deep Neural Network', - 2, - None, - '___sec97'), - ('The final parts of the code', 2, None, '___sec98'), - ('And adding Back propagation', 2, None, '___sec99'), - ('Solving the ODE', 2, None, '___sec100'), - ('Using neural network', 2, None, '___sec101'), - ('Using a deep neural network', 2, None, '___sec102'), - ('Wrapping it up', 2, None, '___sec103')]} + '___sec65')]} end of tocinfo --> @@ -229,7 +171,7 @@ MathJax.Hub.Config({ - Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks + Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning
    @@ -303,44 +245,6 @@ MathJax.Hub.Config({
  • Which activation function should we use?
  • A top-down perspective on Neural networks
  • Limitations of supervised learning with deep networks
    • -
  • Convolutional Neural Networks (recognizing images)
    • -
  • Regular NNs don’t scale well to full images
    • -
  • 3D volumes of neurons
    • -
  • Layers used to build CNNs
    • -
  • Transforming images
    • -
  • CNNs in brief
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Using TensorFlow backend
    • -
  • Train the model
    • -
  • Visualizing the results
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • Fun links
    • -
  • Applications: solving ordinary differential equations with Neural Networks
    • -
  • Trial solution
    • -
  • More details
    • -
  • Reformulating the problem
    • -
  • Estimating errors
    • -
  • Creating a simple Deep Neural Net
    • -
  • Setting up the code, feed forward part
    • -
  • Backpropagation
    • -
  • Gradient Descent
    • -
  • More on GD and cost function
    • -
  • An implementation of a Deep Neural Network
    • -
  • The final parts of the code
    • -
  • And adding Back propagation
    • -
  • Solving the ODE
    • -
  • Using neural network
    • -
  • Using a deep neural network
    • -
  • Wrapping it up
    • @@ -380,6 +284,7 @@ Here we list some of the important limitations of supervised neural network base Some of these remarks are particular to DNNs, others are shared by all supervised learning methods. This motivates the use of unsupervised methods which in part circumvent these problems.

    +

      @@ -395,18 +300,6 @@ Some of these remarks are particular to DNNs, others are shared by all supervise
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    diff --git a/doc/pub/NeuralNet/html/NeuralNet-bs.html b/doc/pub/NeuralNet/html/NeuralNet-bs.html index 9ed465153..2e947261f 100644 --- a/doc/pub/NeuralNet/html/NeuralNet-bs.html +++ b/doc/pub/NeuralNet/html/NeuralNet-bs.html @@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source - + -Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks +Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning @@ -141,65 +141,7 @@ Automatically generated HTML file from DocOnce source ('Limitations of supervised learning with deep networks', 2, None, - '___sec65'), - ('Convolutional Neural Networks (recognizing images)', - 2, - None, - '___sec66'), - ('Regular NNs don’t scale well to full images', - 2, - None, - '___sec67'), - ('3D volumes of neurons', 2, None, '___sec68'), - ('Layers used to build CNNs', 2, None, '___sec69'), - ('Transforming images', 2, None, '___sec70'), - ('CNNs in brief', 2, None, '___sec71'), - ('CNNs in more detail, building convolutional neural networks in ' - 'Tensorflow and Keras', - 2, - None, - '___sec72'), - ('Setting it up', 2, None, '___sec73'), - ('The MNIST dataset again', 2, None, '___sec74'), - ('Strong correlations', 2, None, '___sec75'), - ('Layers of a CNN', 2, None, '___sec76'), - ('Systematic reduction', 2, None, '___sec77'), - ('Prerequisites: Collect and pre-process data', - 2, - None, - '___sec78'), - ('Importing Keras and Tensorflow', 2, None, '___sec79'), - ('Using TensorFlow backend', 2, None, '___sec80'), - ('Train the model', 2, None, '___sec81'), - ('Visualizing the results', 2, None, '___sec82'), - ('Running with Keras', 2, None, '___sec83'), - ('Final part', 2, None, '___sec84'), - ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86'), - ('Applications: solving ordinary differential equations with ' - 'Neural Networks', - 2, - None, - '___sec87'), - ('Trial solution', 2, None, '___sec88'), - ('More details', 2, None, '___sec89'), - ('Reformulating the problem', 2, None, '___sec90'), - ('Estimating errors', 2, None, '___sec91'), - ('Creating a simple Deep Neural Net', 2, None, '___sec92'), - ('Setting up the code, feed forward part', 2, None, '___sec93'), - ('Backpropagation', 2, None, '___sec94'), - ('Gradient Descent', 2, None, '___sec95'), - ('More on GD and cost function', 2, None, '___sec96'), - ('An implementation of a Deep Neural Network', - 2, - None, - '___sec97'), - ('The final parts of the code', 2, None, '___sec98'), - ('And adding Back propagation', 2, None, '___sec99'), - ('Solving the ODE', 2, None, '___sec100'), - ('Using neural network', 2, None, '___sec101'), - ('Using a deep neural network', 2, None, '___sec102'), - ('Wrapping it up', 2, None, '___sec103')]} + '___sec65')]} end of tocinfo --> @@ -229,7 +171,7 @@ MathJax.Hub.Config({ - Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks + Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning
    @@ -303,44 +245,6 @@ MathJax.Hub.Config({
  • Which activation function should we use?
  • A top-down perspective on Neural networks
  • Limitations of supervised learning with deep networks
    • -
  • Convolutional Neural Networks (recognizing images)
    • -
  • Regular NNs don’t scale well to full images
    • -
  • 3D volumes of neurons
    • -
  • Layers used to build CNNs
    • -
  • Transforming images
    • -
  • CNNs in brief
    • -
  • CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
    • -
  • Setting it up
    • -
  • The MNIST dataset again
    • -
  • Strong correlations
    • -
  • Layers of a CNN
    • -
  • Systematic reduction
    • -
  • Prerequisites: Collect and pre-process data
    • -
  • Importing Keras and Tensorflow
    • -
  • Using TensorFlow backend
    • -
  • Train the model
    • -
  • Visualizing the results
    • -
  • Running with Keras
    • -
  • Final part
    • -
  • Final visualization
    • -
  • Fun links
    • -
  • Applications: solving ordinary differential equations with Neural Networks
    • -
  • Trial solution
    • -
  • More details
    • -
  • Reformulating the problem
    • -
  • Estimating errors
    • -
  • Creating a simple Deep Neural Net
    • -
  • Setting up the code, feed forward part
    • -
  • Backpropagation
    • -
  • Gradient Descent
    • -
  • More on GD and cost function
    • -
  • An implementation of a Deep Neural Network
    • -
  • The final parts of the code
    • -
  • And adding Back propagation
    • -
  • Solving the ODE
    • -
  • Using neural network
    • -
  • Using a deep neural network
    • -
  • Wrapping it up
    • @@ -359,7 +263,7 @@ MathJax.Hub.Config({
    -

    Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks

    +

    Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning

    @@ -399,7 +303,7 @@ MathJax.Hub.Config({

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    • diff --git a/doc/pub/NeuralNet/html/NeuralNet-reveal.html b/doc/pub/NeuralNet/html/NeuralNet-reveal.html index acebd4f35..ca21dfb52 100644 --- a/doc/pub/NeuralNet/html/NeuralNet-reveal.html +++ b/doc/pub/NeuralNet/html/NeuralNet-reveal.html @@ -3,9 +3,9 @@ - + -Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks +Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning @@ -132,7 +132,7 @@ MathJax.Hub.Config({ -

    Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks

    +

    Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning

    @@ -3087,1365 +3087,6 @@ Some of these remarks are particular to DNNs, others are shared by all supervise -

    -

    Convolutional Neural Networks (recognizing images)

    - -

    -Convolutional Neural Networks (CNN) are very similar to ordinary Neural Networks. - -

    -They are made up of neurons that have learnable weights and -biases. Each neuron receives some inputs, performs a dot product and -optionally follows it with a non-linearity. The whole network still -expresses a single differentiable score function: from the raw image -pixels on one end to class scores at the other. And they still have a -loss function (for example Softmax) on the last (fully-connected) layer -and all the tips/tricks we developed for learning regular Neural -Networks still apply (back propagation, gradient descent etc etc). - -

    -What is the difference? CNN architectures make the explicit assumption that -the inputs are images, which allows us to encode certain properties -into the architecture. These then make the forward function more -efficient to implement and vastly reduce the amount of parameters in -the network. - -

    -Here we provide only a superficial overview, for the more interested, we recommend highly the course -IN5400 – Machine Learning for Image Analysis -and the slides of CS231. -

    - - -
    -

    Regular NNs don’t scale well to full images

    - -

    -As an example, consider -an image of size \( 32\times 32\times 3 \) (32 wide, 32 high, 3 color channels), so a -single fully-connected neuron in a first hidden layer of a regular -Neural Network would have \( 32\times 32\times 3 = 3072 \) weights. This amount still -seems manageable, but clearly this fully-connected structure does not -scale to larger images. For example, an image of more respectable -size, say \( 200\times 200\times 3 \), would lead to neurons that have -\( 200\times 200\times 3 = 120,000 \) weights. - -

    -We could have -several such neurons, and the parameters would add up quickly! Clearly, -this full connectivity is wasteful and the huge number of parameters -would quickly lead to possible overfitting. - -

    -

    -
    -

    Figure 1: A regular 3-layer Neural Network.

    -

    -
    -
    - - -
    -

    3D volumes of neurons

    - -

    -Convolutional Neural Networks take advantage of the fact that the -input consists of images and they constrain the architecture in a more -sensible way. - -

    -In particular, unlike a regular Neural Network, the -layers of a CNN have neurons arranged in 3 dimensions: width, -height, depth. (Note that the word depth here refers to the third -dimension of an activation volume, not to the depth of a full Neural -Network, which can refer to the total number of layers in a network.) - -

    -To understand it better, the above example of an image -with an input volume of -activations has dimensions \( 32\times 32\times 3 \) (width, height, -depth respectively). - -

    -The neurons in a layer will -only be connected to a small region of the layer before it, instead of -all of the neurons in a fully-connected manner. Moreover, the final -output layer could for this specific image have dimensions \( 1\times 1 \times 10 \), -because by the -end of the CNN architecture we will reduce the full image into a -single vector of class scores, arranged along the depth -dimension. - -

    -

    -
    -

    Figure 2: A CNN arranges its neurons in three dimensions (width, height, depth), as visualized in one of the layers. Every layer of a CNN transforms the 3D input volume to a 3D output volume of neuron activations. In this example, the red input layer holds the image, so its width and height would be the dimensions of the image, and the depth would be 3 (Red, Green, Blue channels).

    -

    -
    -
    - - -
    -

    Layers used to build CNNs

    - -

    -A simple CNN is a sequence of layers, and every layer of a CNN -transforms one volume of activations to another through a -differentiable function. We use three main types of layers to build -CNN architectures: Convolutional Layer, Pooling Layer, and -Fully-Connected Layer (exactly as seen in regular Neural Networks). We -will stack these layers to form a full CNN architecture. - -

    -A simple CNN for image classification could have the architecture: - -

      -

    • INPUT (\( 32\times 32 \times 3 \)) will hold the raw pixel values of the image, in this case an image of width 32, height 32, and with three color channels R,G,B.
      • -

    • CONV (convolutional )layer will compute the output of neurons that are connected to local regions in the input, each computing a dot product between their weights and a small region they are connected to in the input volume. This may result in volume such as \( [32\times 32\times 12] \) if we decided to use 12 filters.
      • -

    • RELU layer will apply an elementwise activation function, such as the \( max(0,x) \) thresholding at zero. This leaves the size of the volume unchanged (\( [32\times 32\times 12] \)).
      • -

    • POOL (pooling) layer will perform a downsampling operation along the spatial dimensions (width, height), resulting in volume such as \( [16\times 16\times 12] \).
      • -

    • FC (i.e. fully-connected) layer will compute the class scores, resulting in volume of size \( [1\times 1\times 10] \), where each of the 10 numbers correspond to a class score, such as among the 10 categories of the MNIST images we considered above . As with ordinary Neural Networks and as the name implies, each neuron in this layer will be connected to all the numbers in the previous volume.
      • -
    -
    - - -
    -

    Transforming images

    - -

    -CNNs transform the original image layer by layer from the original -pixel values to the final class scores. - -

    -Observe that some layers contain -parameters and other don’t. In particular, the CNN layers perform -transformations that are a function of not only the activations in the -input volume, but also of the parameters (the weights and biases of -the neurons). On the other hand, the RELU/POOL layers will implement a -fixed function. The parameters in the CONV/FC layers will be trained -with gradient descent so that the class scores that the CNN computes -are consistent with the labels in the training set for each image. -

    - - -
    -

    CNNs in brief

    - -

    -In summary: - -

      -

    • A CNN architecture is in the simplest case a list of Layers that transform the image volume into an output volume (e.g. holding the class scores)
      • -

    • There are a few distinct types of Layers (e.g. CONV/FC/RELU/POOL are by far the most popular)
      • -

    • Each Layer accepts an input 3D volume and transforms it to an output 3D volume through a differentiable function
      • -

    • Each Layer may or may not have parameters (e.g. CONV/FC do, RELU/POOL don’t)
      • -

    • Each Layer may or may not have additional hyperparameters (e.g. CONV/FC/POOL do, RELU doesn’t)
      • -
    -

    - -For more material on convolutional networks, we strongly recommend -the course -IN5400 – Machine Learning for Image Analysis -and the slides of CS231 which is taught at Stanford University (consistently ranked as one of the top computer science programs in the world). Michael Nielsen's book is a must read, in particular chapter 6 which deals with CNNs. -

    - - -
    -

    CNNs in more detail, building convolutional neural networks in Tensorflow and Keras

    - -

    -As discussed above, CNNs are neural networks built from the assumption that the inputs -to the network are 2D images. This is important because the number of features or pixels in images -grows very fast with the image size, and an enormous number of weights and biases are needed in order to build an accurate network. - -

    -As before, we still have our input, a hidden layer and an output. What's novel about convolutional networks -are the convolutional and pooling layers stacked in pairs between the input and the hidden layer. -In addition, the data is no longer represented as a 2D feature matrix, instead each input is a number of 2D -matrices, typically 1 for each color dimension (Red, Green, Blue). -

    - - -
    -

    Setting it up

    - -

    -It means that to represent the entire -dataset of images, we require a 4D matrix or tensor. This tensor has the dimensions: - -

     
    -$$ -(n_{inputs},\, n_{pixels, width},\, n_{pixels, height},\, depth) . -$$ -

     
    -

    - - -
    -

    The MNIST dataset again

    - -

    -The MNIST dataset consists of grayscale images with a pixel size of -\( 28\times 28 \), meaning we require \( 28 \times 28 = 724 \) weights to each -neuron in the first hidden layer. - -

    -If we were to analyze images of size \( 128\times 128 \) we would require -\( 128 \times 128 = 16384 \) weights to each neuron. Even worse if we were -dealing with color images, as most images are, we have an image matrix -of size \( 128\times 128 \) for each color dimension (Red, Green, Blue), -meaning 3 times the number of weights \( = 49152 \) are required for every -single neuron in the first hidden layer. -

    - - -
    -

    Strong correlations

    -Images typically have strong local correlations, meaning that a small -part of the image varies little from its neighboring regions. If for -example we have an image of a blue car, we can roughly assume that a -small blue part of the image is surrounded by other blue regions. - -

    -Therefore, instead of connecting every single pixel to a neuron in the -first hidden layer, as we have previously done with deep neural -networks, we can instead connect each neuron to a small part of the -image (in all 3 RGB depth dimensions). The size of each small area is -fixed, and known as a receptive. -

    - - -
    -

    Layers of a CNN

    -The layers of a convolutional neural network arrange neurons in 3D: width, height and depth. -The input image is typically a square matrix of depth 3. - -

    -A convolution is performed on the image which outputs -a 3D volume of neurons. The weights to the input are arranged in a number of 2D matrices, known as filters. - -

    -Each filter slides along the input image, taking the dot product -between each small part of the image and the filter, in all depth -dimensions. This is then passed through a non-linear function, -typically the Rectified Linear (ReLu) function, which serves as the -activation of the neurons in the first convolutional layer. This is -further passed through a pooling layer, which reduces the size of the -convolutional layer, e.g. by taking the maximum or average across some -small regions, and this serves as input to the next convolutional -layer. -

    - - -
    -

    Systematic reduction

    - -

    -By systematically reducing the size of the input volume, through -convolution and pooling, the network should create representations of -small parts of the input, and then from them assemble representations -of larger areas. The final pooling layer is flattened to serve as -input to a hidden layer, such that each neuron in the final pooling -layer is connected to every single neuron in the hidden layer. This -then serves as input to the output layer, e.g. a softmax output for -classification. -

    - - -
    -

    Prerequisites: Collect and pre-process data

    -

    - - -

    # import necessary packages
-import numpy as np
-import matplotlib.pyplot as plt
-from sklearn import datasets
-
-
-# ensure the same random numbers appear every time
-np.random.seed(0)
-
-# display images in notebook
-%matplotlib inline
-plt.rcParams['figure.figsize'] = (12,12)
-
-
-# download MNIST dataset
-digits = datasets.load_digits()
-
-# define inputs and labels
-inputs = digits.images
-labels = digits.target
-
-# RGB images have a depth of 3
-# our images are grayscale so they should have a depth of 1
-inputs = inputs[:,:,:,np.newaxis]
-
-print("inputs = (n_inputs, pixel_width, pixel_height, depth) = " + str(inputs.shape))
-print("labels = (n_inputs) = " + str(labels.shape))
-
-
-# choose some random images to display
-n_inputs = len(inputs)
-indices = np.arange(n_inputs)
-random_indices = np.random.choice(indices, size=5)
-
-for i, image in enumerate(digits.images[random_indices]):
-    plt.subplot(1, 5, i+1)
-    plt.axis('off')
-    plt.imshow(image, cmap=plt.cm.gray_r, interpolation='nearest')
-    plt.title("Label: %d" % digits.target[random_indices[i]])
-plt.show()
-
    -
    - - -
    -

    Importing Keras and Tensorflow

    -

    - - -

    from keras.utils import to_categorical
-from sklearn.model_selection import train_test_split
-
-# representation of labels
-labels = to_categorical(labels)
-
-# split into train and test data
-# one-liner from scikit-learn library
-train_size = 0.8
-test_size = 1 - train_size
-X_train, X_test, Y_train, Y_test = train_test_split(inputs, labels, train_size=train_size,
-                                                    test_size=test_size)
-
    -
    - - -
    -

    Using TensorFlow backend

    - -

    -We need to define model and architecture and choose cost function and optmizer. -

    - - -

    import tensorflow as tf
-
-class ConvolutionalNeuralNetworkTensorflow:
-    def __init__(
-            self,
-            X_train,
-            Y_train,
-            X_test,
-            Y_test,
-            n_filters=10,
-            n_neurons_connected=50,
-            n_categories=10,
-            receptive_field=3,
-            stride=1,
-            padding=1,
-            epochs=10,
-            batch_size=100,
-            eta=0.1,
-            lmbd=0.0):
-        
-        self.global_step = tf.Variable(0, dtype=tf.int32, trainable=False, name='global_step')
-        
-        self.X_train = X_train
-        self.Y_train = Y_train
-        self.X_test = X_test
-        self.Y_test = Y_test
-        
-        self.n_inputs, self.input_width, self.input_height, self.depth = X_train.shape
-        
-        self.n_filters = n_filters
-        self.n_downsampled = int(self.input_width*self.input_height*n_filters / 4)
-        self.n_neurons_connected = n_neurons_connected
-        self.n_categories = n_categories
-        
-        self.receptive_field = receptive_field
-        self.stride = stride
-        self.strides = [stride, stride, stride, stride]
-        self.padding = padding
-        
-        self.epochs = epochs
-        self.batch_size = batch_size
-        self.iterations = self.n_inputs // self.batch_size
-        self.eta = eta
-        self.lmbd = lmbd
-        
-        self.create_placeholders()
-        self.create_CNN()
-        self.create_loss()
-        self.create_optimiser()
-        self.create_accuracy()
-    
-    def create_placeholders(self):
-        with tf.name_scope('data'):
-            self.X = tf.placeholder(tf.float32, shape=(None, self.input_width, self.input_height, self.depth), name='X_data')
-            self.Y = tf.placeholder(tf.float32, shape=(None, self.n_categories), name='Y_data')
-    
-    def create_CNN(self):
-        with tf.name_scope('CNN'):
-            
-            # Convolutional layer
-            self.W_conv = self.weight_variable([self.receptive_field, self.receptive_field, self.depth, self.n_filters], name='conv', dtype=tf.float32)
-            b_conv = self.weight_variable([self.n_filters], name='conv', dtype=tf.float32)
-            z_conv = tf.nn.conv2d(self.X, self.W_conv, self.strides, padding='SAME', name='conv') + b_conv
-            a_conv = tf.nn.relu(z_conv)
-            
-            # 2x2 max pooling
-            a_pool = tf.nn.max_pool(a_conv, [1, 2, 2, 1], [1, 2, 2, 1], padding='SAME', name='pool')
-            
-            # Fully connected layer
-            a_pool_flat = tf.reshape(a_pool, [-1, self.n_downsampled])
-            self.W_fc = self.weight_variable([self.n_downsampled, self.n_neurons_connected], name='fc', dtype=tf.float32)
-            b_fc = self.bias_variable([self.n_neurons_connected], name='fc', dtype=tf.float32)
-            a_fc = tf.nn.relu(tf.matmul(a_pool_flat, self.W_fc) + b_fc)
-            
-            # Output layer
-            self.W_out = self.weight_variable([self.n_neurons_connected, self.n_categories], name='out', dtype=tf.float32)
-            b_out = self.bias_variable([self.n_categories], name='out', dtype=tf.float32)
-            self.z_out = tf.matmul(a_fc, self.W_out) + b_out
-    
-    def create_loss(self):
-        with tf.name_scope('loss'):
-            softmax_loss = tf.reduce_mean(tf.nn.softmax_cross_entropy_with_logits_v2(labels=self.Y, logits=self.z_out))
-            
-            regularizer_loss_conv = tf.nn.l2_loss(self.W_conv)
-            regularizer_loss_fc = tf.nn.l2_loss(self.W_fc)
-            regularizer_loss_out = tf.nn.l2_loss(self.W_out)
-            regularizer_loss = self.lmbd*(regularizer_loss_conv + regularizer_loss_fc + regularizer_loss_out)
-            
-            self.loss = softmax_loss + regularizer_loss
-
-    def create_accuracy(self):
-        with tf.name_scope('accuracy'):
-            probabilities = tf.nn.softmax(self.z_out)
-            predictions = tf.argmax(probabilities, 1)
-            labels = tf.argmax(self.Y, 1)
-            
-            correct_predictions = tf.equal(predictions, labels)
-            correct_predictions = tf.cast(correct_predictions, tf.float32)
-            self.accuracy = tf.reduce_mean(correct_predictions)
-    
-    def create_optimiser(self):
-        with tf.name_scope('optimizer'):
-            self.optimizer = tf.train.GradientDescentOptimizer(learning_rate=self.eta).minimize(self.loss, global_step=self.global_step)
-            
-    def weight_variable(self, shape, name='', dtype=tf.float32):
-        initial = tf.truncated_normal(shape, stddev=0.1)
-        return tf.Variable(initial, name=name, dtype=dtype)
-    
-    def bias_variable(self, shape, name='', dtype=tf.float32):
-        initial = tf.constant(0.1, shape=shape)
-        return tf.Variable(initial, name=name, dtype=dtype)
-
-    def fit(self):
-        data_indices = np.arange(self.n_inputs)
-
-        with tf.Session() as sess:
-            sess.run(tf.global_variables_initializer())
-            for i in range(self.epochs):
-                for j in range(self.iterations):
-                    chosen_datapoints = np.random.choice(data_indices, size=self.batch_size, replace=False)
-                    batch_X, batch_Y = self.X_train[chosen_datapoints], self.Y_train[chosen_datapoints]
-            
-                    sess.run([CNN.loss, CNN.optimizer],
-                        feed_dict={CNN.X: batch_X,
-                                   CNN.Y: batch_Y})
-                    accuracy = sess.run(CNN.accuracy,
-                        feed_dict={CNN.X: batch_X,
-                                   CNN.Y: batch_Y})
-                    step = sess.run(CNN.global_step)
-    
-            self.train_loss, self.train_accuracy = sess.run([CNN.loss, CNN.accuracy],
-                feed_dict={CNN.X: self.X_train,
-                           CNN.Y: self.Y_train})
-        
-            self.test_loss, self.test_accuracy = sess.run([CNN.loss, CNN.accuracy],
-                feed_dict={CNN.X: self.X_test,
-                           CNN.Y: self.Y_test})
-
    -
    - - -
    -

    Train the model

    - -

    -We need now to train the model, evaluate it and test its performance on test data, and eventually include hyperparameters. -

    - - -

    epochs = 100
-batch_size = 100
-n_filters = 10
-n_neurons_connected = 50
-n_categories = 10
-
-eta_vals = np.logspace(-5, 1, 7)
-lmbd_vals = np.logspace(-5, 1, 7)
-CNN_tf = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)
-        
-for i, eta in enumerate(eta_vals):
-    for j, lmbd in enumerate(lmbd_vals):
-        CNN = ConvolutionalNeuralNetworkTensorflow(X_train, Y_train, X_test, Y_test,
-                                      n_filters=n_filters, n_neurons_connected=n_neurons_connected,
-                                      n_categories=n_categories, epochs=epochs, batch_size=batch_size,
-                                      eta=eta, lmbd=lmbd)
-        CNN.fit()
-        
-        print("Learning rate = ", eta)
-        print("Lambda = ", lmbd)
-        print("Test accuracy: %.3f" % CNN.test_accuracy)
-        print()
-            
-        CNN_tf[i][j] = CNN
-
    -
    - - -
    -

    Visualizing the results

    - -

    - - -

    # visual representation of grid search
-# uses seaborn heatmap, could probably do this in matplotlib
-import seaborn as sns
-
-sns.set()
-
-train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
-test_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
-
-for i in range(len(eta_vals)):
-    for j in range(len(lmbd_vals)):
-        CNN = CNN_tf[i][j]
-
-        train_accuracy[i][j] = CNN.train_accuracy
-        test_accuracy[i][j] = CNN.test_accuracy
-
-        
-fig, ax = plt.subplots(figsize = (10, 10))
-sns.heatmap(train_accuracy, annot=True, ax=ax, cmap="viridis")
-ax.set_title("Training Accuracy")
-ax.set_ylabel("$\eta$")
-ax.set_xlabel("$\lambda$")
-plt.show()
-
-fig, ax = plt.subplots(figsize = (10, 10))
-sns.heatmap(test_accuracy, annot=True, ax=ax, cmap="viridis")
-ax.set_title("Test Accuracy")
-ax.set_ylabel("$\eta$")
-ax.set_xlabel("$\lambda$")
-plt.show()
-
    -
    - - -
    -

    Running with Keras

    - -

    - - -

    from keras.models import Sequential
-from keras.layers.convolutional import Conv2D
-from keras.layers.convolutional import MaxPooling2D
-from keras.layers import Flatten
-from keras.layers import Dense
-from keras.regularizers import l2
-from keras.optimizers import SGD
-
-def create_convolutional_neural_network_keras(input_shape, receptive_field,
-                                              n_filters, n_neurons_connected, n_categories,
-                                              eta, lmbd):
-    model = Sequential()
-    model.add(Conv2D(n_filters, (receptive_field, receptive_field), input_shape=input_shape, padding='same',
-              activation='relu', kernel_regularizer=l2(lmbd)))
-    model.add(MaxPooling2D(pool_size=(2, 2)))
-    model.add(Flatten())
-    model.add(Dense(n_neurons_connected, activation='relu', kernel_regularizer=l2(lmbd)))
-    model.add(Dense(n_categories, activation='softmax', kernel_regularizer=l2(lmbd)))
-    
-    sgd = SGD(lr=eta)
-    model.compile(loss='categorical_crossentropy', optimizer=sgd, metrics=['accuracy'])
-    
-    return model
-
-epochs = 100
-batch_size = 100
-input_shape = X_train.shape[1:4]
-receptive_field = 3
-n_filters = 10
-n_neurons_connected = 50
-n_categories = 10
-
-eta_vals = np.logspace(-5, 1, 7)
-lmbd_vals = np.logspace(-5, 1, 7)
-
    -
    - - -
    -

    Final part

    - -

    - - -

    CNN_keras = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)
-        
-for i, eta in enumerate(eta_vals):
-    for j, lmbd in enumerate(lmbd_vals):
-        CNN = create_convolutional_neural_network_keras(input_shape, receptive_field,
-                                              n_filters, n_neurons_connected, n_categories,
-                                              eta, lmbd)
-        CNN.fit(X_train, Y_train, epochs=epochs, batch_size=batch_size, verbose=0)
-        scores = CNN.evaluate(X_test, Y_test)
-        
-        CNN_keras[i][j] = CNN
-        
-        print("Learning rate = ", eta)
-        print("Lambda = ", lmbd)
-        print("Test accuracy: %.3f" % scores[1])
-        print()
-
    -
    - - -
    -

    Final visualization

    - -

    - - -

    # visual representation of grid search
-# uses seaborn heatmap, could probably do this in matplotlib
-import seaborn as sns
-
-sns.set()
-
-train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
-test_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
-
-for i in range(len(eta_vals)):
-    for j in range(len(lmbd_vals)):
-        CNN = CNN_keras[i][j]
-
-        train_accuracy[i][j] = CNN.evaluate(X_train, Y_train)[1]
-        test_accuracy[i][j] = CNN.evaluate(X_test, Y_test)[1]
-
-        
-fig, ax = plt.subplots(figsize = (10, 10))
-sns.heatmap(train_accuracy, annot=True, ax=ax, cmap="viridis")
-ax.set_title("Training Accuracy")
-ax.set_ylabel("$\eta$")
-ax.set_xlabel("$\lambda$")
-plt.show()
-
-fig, ax = plt.subplots(figsize = (10, 10))
-sns.heatmap(test_accuracy, annot=True, ax=ax, cmap="viridis")
-ax.set_title("Test Accuracy")
-ax.set_ylabel("$\eta$")
-ax.set_xlabel("$\lambda$")
-plt.show()
-
    -
    - - -
    -

    Fun links

    - -
      -

    1. Self-Driving cars using a convolutional neural network
      2. -

    2. Abstract art using convolutional neural networks
      3. -
    -
    - - -
    -

    Applications: solving ordinary differential equations with Neural Networks

    - -

    -We end our discussion on neural networks with a discussion on how to solve differential equations. Here we focus -first on the classical exponential decay in one dimension. Thereafter we switch to the Poisson equation in one dimension. - -

    -The aim is to see if we can use a neural network to solve - -

     
    -$$ -\begin{equation} -\tag{16} -g'(x) = -\gamma g(x) -\end{equation} -$$ -

     
    - -

    -where \( g(0) = g_0 \) with \( \gamma \) and \( g_0 \) being some chosen -values. This equation is an ordinary differential equation since the -function we have to solve for, \( g(x) \), is of one variable. - -

    -Here we set \( \gamma = 2 \) and \( g_0 = 10 \) but feel free to change -them and see how the neural network performs. -

    - - -
    -

    Trial solution

    - -

    -To begin with, a trial solution \( g_t(t) \) must be chosen. A general -trial solution for ordinary differential equations could be - -

     
    -$$ -g_t(x, P) = h_1(x) + h_2(x, N(x, P)), -$$ -

     
    - -

    -with \( h_1(x) \) ensuring that \( g_t(x) \) satisfies some conditions and -\( h_2(x,N(x, P)) \) an expression involving \( x \) and the output from the -neural network \( N(x,P) \) with \( P \) being the collection of the weights -and biases for each layer. - -

    -It is assumed that there are no weights and -bias at the input layer, so \( P = \{ P_{\mathrm{hidden}}, -P_{\mathrm{output}} \} \). If there are \( N_{\mathrm{hidden} } \) neurons in -the hidden layer, then \( P_{\mathrm{hidden}} \) is an \( N_{\mathrm{hidden} } -\times 2 \) matrix. - -

    -The first column in \( P_{\mathrm{hidden} } \) represents -the bias for each neuron in the hidden layer and the second column -represents the weigths for each neuron. If there are \( N_{\mathrm{output} -} \) neurons in the output layer, then \( P_{\mathrm{output}} \) is a -\( N_{\mathrm{output} } \times (1 + N_{\mathrm{hidden} }) \) matrix. Its first -column represents the bias of each neuron and the remaining columns -represents the weights to each neuron. -

    - - -
    -

    More details

    - -

    -We have \( g(0) = g_0 \). The trial solution must fulfill this -condition to be a proper solution of (16). - -

    -A possible way to -ensure that \( g_t(0, P) = g_0 \), is to let \( F(N(x,P)) = x\cdot N(x,P) \) -and \( A(x) = g_0 \). This gives the following trial solution: - -

     
    -$$ -\begin{equation} -g_t(x, P) = g_0 + x \cdot N(x, P). -\tag{17} -\end{equation} -$$ -

     
    -

    - - -
    -

    Reformulating the problem

    - -

    -Often, the role of a neural network is to minimize its parameters with -respect to some given error criteria. This criteria, the cost or loss -function, is a measure of how much error the output of the network has -compared to some given known answers. A reformulation of -(16) must therefore be done, such that it describes the -problem a neural network can solve. - -

    -The neural network must find the set of weigths and biases \( P \) such -that the trial solution in satisfies -(16). The trial solution has been chosen such that it -already solves the condition \( g(0) = g_0 \). What remains, is to find -\( P \) such that - -

     
    -$$ -\begin{equation} -g_t'(x, P) = - \gamma g_t(x, P) -\tag{18} -\end{equation} -$$ -

     
    - -

    -is fulfilled as best as possible. -

    - - -
    -

    Estimating errors

    - -

    -Having two sides of an equation as equal as -possible, means that the absolute or squared difference between the -sides must be as close to zero as small. In this case, the difference -squared is an appropiate measurement of how errorneous the trial -solution is with respect to \( P \) of the neural network. Therefore, the -problem our network must solve, is - -

     
    -$$ -\min_{P}\Big\{ \big(g_t'(x, P) - ( -\gamma g_t(x, P) \big)^2 \Big\} -$$ -

     
    - -

    -or, in terms of weights and biases for each layer: - -

     
    -$$ -\min_{P_{\mathrm{hidden} }, \ P_{\mathrm{output} }}\Big\{ \big(g_t'(x, \{ P_{\mathrm{hidden} }, P_{\mathrm{output} }\}) - ( -\gamma g_t(x, \{ P_{\mathrm{hidden} }, P_{\mathrm{output} }\}) \big)^2 \Big\} -$$ -

     
    - -

    -for an input value \( x \). -If the neural network evaluates \( g_t(x, P) \) at more avalues for \( x \), say \( N \) values \( x_i \) for \( i = 1, \dots, N \), then the total error to minimize is - -

     
    -$$ -\begin{equation} -\tag{19} -\min_{P}\Big\{\sum_i \big(g_t'(x_i, P) - ( -\gamma g_t(x_i, P) \big)^2 \Big\} -\end{equation} -$$ -

     
    - -

    -Letting \( c(x, P) = \sum_i \big(g_t'(x_i, P) - ( -\gamma g_t(x_i, P) -\big)^2 \) denote the cost function, the minimization problem of which -our network must solve, is - -

     
    -$$ -\min_{P} c(x, P) -$$ -

     
    - -

    -or in terms of \( P_{\mathrm{hidden} } \) and \( P_{\mathrm{output} } \) - -

     
    -$$ -\min_{P_{\mathrm{hidden} }, \ P_{\mathrm{output} }} c(x, \{P_{\mathrm{hidden} }, P_{\mathrm{output} }\}) -$$ -

     
    -

    - - -
    -

    Creating a simple Deep Neural Net

    - -

    -The next step is to decide how the neural net \( N(x, P) \) -should be. In this case, the neural network is made -from scratch to understand better how a neural network works, gain -more control over its architecture, and see how Autograd can be used -to simplify the implementation. - -

    -Since a deep neural network (DNN) is a neural network with more than -one hidden layer, we can first look on how to implement a neural -network. Having an implementation of a neural network at hand, an -extension of it into a deep neural network would (hopefully) be -painless. - -

    -For simplicity, we assume that the input is an array -\( \hat{x}= (x_1, \dots, x_N) \) with \( N \) elements. It is at these points the neural -network should find \( P \) such that it fulfills (19). -All the ingredients discussed earlier, from the activation function, hidden layers and their weights, biases etc -are included below. -

    - - -
    -

    Setting up the code, feed forward part

    -

    - - -

    # Note that we use the  numpy wrapper for Autograd (see the gradient descent  slides)
-import autograd.numpy as np
-from autograd import grad, elementwise_grad
-import autograd.numpy.random as npr
-from matplotlib import pyplot as plt
-
-def sigmoid(z):
-    return 1/(1 + np.exp(-z))
-
-def neural_network(params, x):
-    
-    # Find the weights (including and biases) for the hidden and output layer.
-    # Assume that params is a list of parameters for each layer. 
-    # The biases are the first element for each array in params, 
-    # and the weights are the remaning elements in each array in params.   
-    
-    w_hidden = params[0]
-    w_output = params[1]
-
-    # Assumes input x being an one-dimensional array
-    num_values = np.size(x)
-    x = x.reshape(-1, num_values)
-    
-    # Assume that the input layer does nothing to the input x
-    x_input = x
-
-    ## Hidden layer:
-    
-    # Add a row of ones to include bias
-    x_input = np.concatenate((np.ones((1,num_values)), x_input ), axis = 0)
-    
-    z_hidden = np.matmul(w_hidden, x_input)
-    x_hidden = sigmoid(z_hidden)
-
-    ## Output layer:
-    
-    # Include bias:
-    x_hidden = np.concatenate((np.ones((1,num_values)), x_hidden ), axis = 0)
-
-    z_output = np.matmul(w_output, x_hidden)
-    x_output = z_output
-
-    return x_output
-
    -
    - - -
    -

    Backpropagation

    - -

    -Now that the feedforward can be done, the next step is to decide how the -parameters should change such that they minimize the cost function. - -

    -Recall that the chosen cost function for this problem is - -

     
    -$$ -c(x, P) = \sum_i \big(g_t'(x_i, P) - ( -\gamma g_t(x_i, P) \big)^2 -$$ -

     
    - -

    -In order to minimize it, an optimization method must be chosen. - -

    -Here, gradient descent with a constant step size has been chosen. - -

    -Before looking at the gradient descent method, let us set up the cost -function along with the right ride of the ODE and trial solution. - -

    - - -

    # The trial solution using the deep neural network:
-def g_trial(x,params, g0 = 10):
-    return g0 + x*neural_network(params,x)
-
-# The right side of the ODE:
-def g(x, g_trial, gamma = 2):
-    return -gamma*g_trial
-
-# The cost function:
-def cost_function(P, x):
-    
-    # Evaluate the trial function with the current parameters P
-    g_t = g_trial(x,P)
-    
-    # Find the derivative w.r.t x of the neural network
-    d_net_out = elementwise_grad(neural_network,1)(P,x) 
-    
-    # Find the derivative w.r.t x of the trial function
-    d_g_t = elementwise_grad(g_trial,0)(x,P)  
-    
-    # The right side of the ODE 
-    func = g(x, g_t)
-
-    err_sqr = (d_g_t - func)**2
-    cost_sum = np.sum(err_sqr)
-    
-    return cost_sum
-
    -
    - - -
    -

    Gradient Descent

    - -

    -The idea of the gradient descent algorithm is to update parameters in -direction where the cost function decreases goes to a minimum. - -

    -In general, the update of some parameters \( \hat{\omega} \) given a cost -function defined by some weights \( \hat{\omega} \), \( c(x, \hat{\omega}) \), -goes as follows: - -

     
    -$$ -\hat{\omega}_{\mathrm{new} } = \hat{\omega} - \lambda \nabla_{\hat{\omega}} c(x, \hat{\omega}), -$$ -

     
    - -

    -for a number of iterations or until \( \big|\big| \hat{\omega}_{\mathrm{new} } - \hat{\omega} \big|\big| \) -is smaller than some -given tolerance. - -

    -The value of \( \lambda \) decides how large steps the algorithm must take -in the direction of $ \nabla_{\hat{\omega}} c(x, \hat{\omega})$. The -notatation \( \nabla_{\hat{\omega}} \) denotes the gradient with respect to -the elements in \( \hat{\omega} \). -

    - - -
    -

    More on GD and cost function

    - -

    -In our case, we have to minimize the cost function \( c(x, P) \) with -respect to the two sets of weights and bisases, that is for the hidden -layer \( P_{\mathrm{hidden} } \) and for the ouput layer \( P_{\mathrm{output} -} \) . - -

    -This means that \( P_{\mathrm{hidden} } \) and \( P_{\mathrm{output} } \) is -updated by - -

     
    -$$ -\begin{align} -P_{\mathrm{hidden},\mathrm{new}} &= P_{\mathrm{hidden}} - \lambda \nabla_{P_{\mathrm{hidden}}} c(x, P) -\tag{20}\\ -P_{\mathrm{output},\mathrm{new}} &= P_{\mathrm{output}} - \lambda \nabla_{P_{\mathrm{output}}} c(x, P) -\tag{21} -\end{align} -$$ -

     
    - -

    -This might look like a cumberstone to set up the correct expression -for finding the gradients. Luckily, Autograd comes to the rescue. - -

    - - -

    def solve_ode_neural_network(x, num_neurons_hidden, num_iter, lmb):
-    ## Set up initial weigths and biases 
-    
-    # For the hidden layer
-    p0 = npr.randn(num_neurons_hidden, 2 ) 
-
-    # For the output layer
-    p1 = npr.randn(1, num_neurons_hidden + 1 ) # +1 since bias is included
-
-    P = [p0, p1]
-
-    print('Initial cost: %g'%cost_function(P, x))
-    
-    ## Start finding the optimal weigths using gradient descent
-    
-    # Find the Python function that represents the gradient of the cost function
-    # w.r.t the 0-th input argument -- that is the weights and biases in the hidden and output layer
-    cost_function_grad = grad(cost_function,0)
-    
-    # Let the update be done num_iter times
-    for i in range(num_iter):
-        # Evaluate the gradient at the current weights and biases in P. 
-        # The cost_grad consist now of two arrays; 
-        # one for the gradient w.r.t P_hidden and 
-        # one for the gradient w.r.t P_output
-        cost_grad =  cost_function_grad(P, x)
-    
-        P[0] = P[0] - lmb * cost_grad[0]
-        P[1] = P[1] - lmb * cost_grad[1]
-
-    print('Final cost: %g'%cost_function(P, x))
-    
-    return P
-
    -
    - - -
    -

    An implementation of a Deep Neural Network

    - -

    -As previously stated, a Deep Neural Network (DNN) follows the same -concept of a neural network, but having more than one hidden -layer. Suppose that the network has \( N_{\mathrm{hidden}} \) hidden layers -where the \( l \)-th layer has \( N_{\mathrm{hidden}}^{(l)} \) neurons. The -input is still assumed to be an array of size \( 1 \times N \). The -network must now try to optimalize its output with respect to the -collection of weigths and biases \( P = \big\{P_{\mathrm{input} }, \ -P_{\mathrm{hidden} }^{(1)}, \ P_{\mathrm{hidden} }^{(2)}, \ \dots , \ -P_{\mathrm{hidden} }^{(N_{\mathrm{hidden}})}, \ P_{\mathrm{output} }\big\} \). -

    - - -
    -

    The final parts of the code

    -

    - - -

    def deep_neural_network(deep_params, x):
-    # N_hidden is the number of hidden layers  
-    N_hidden = np.size(deep_params) - 1 # -1 since params consist of parameters to all the hidden layers AND the output layer
-        
-    # Assumes input x being an one-dimensional array
-    num_values = np.size(x)
-    x = x.reshape(-1, num_values)
-    
-    # Assume that the input layer does nothing to the input x
-    x_input = x
-    
-    # Due to multiple hidden layers, define a variable referencing to the
-    # output of the previous layer:
-    x_prev = x_input 
-    
-    ## Hidden layers:
-    
-    for l in range(N_hidden):
-        # From the list of parameters P; find the correct weigths and bias for this layer
-        w_hidden = deep_params[l]
-        
-        # Add a row of ones to include bias
-        x_prev = np.concatenate((np.ones((1,num_values)), x_prev ), axis = 0)
-
-        z_hidden = np.matmul(w_hidden, x_prev)
-        x_hidden = sigmoid(z_hidden)
-
-        # Update x_prev such that next layer can use the output from this layer
-        x_prev = x_hidden 
-
-    ## Output layer:
-    
-    # Get the weights and bias for this layer
-    w_output = deep_params[-1]
-    
-    # Include bias:
-    x_prev = np.concatenate((np.ones((1,num_values)), x_prev), axis = 0)
-
-    z_output = np.matmul(w_output, x_prev)
-    x_output = z_output
-
-    return x_output
-
    -
    - - -
    -

    And adding Back propagation

    - -

    -This step is very similar for the neural network. The idea in this -step is the same as for the neural network, but with more parameters -to update for. Again there is no need for computing the gradients -analytically since Autograd does the work for us. - -

    - - -

    # The trial solution using the deep neural network:
-def g_trial_deep(x,params, g0 = 10):
-    return g0 + x*deep_neural_network(params,x)
-
-# The same cost function as for the neural network, but calls deep_neural_network instead.
-def cost_function_deep(P, x):
-    
-    # Evaluate the trial function with the current parameters P
-    g_t = g_trial_deep(x,P)
-    
-    # Find the derivative w.r.t x of the neural network
-    d_net_out = elementwise_grad(deep_neural_network,1)(P,x) 
-    
-    # Find the derivative w.r.t x of the trial function
-    d_g_t = elementwise_grad(g_trial_deep,0)(x,P)  
-    
-    # The right side of the ODE 
-    func = g(x, g_t)
-
-    err_sqr = (d_g_t - func)**2
-    cost_sum = np.sum(err_sqr)
-    
-    return cost_sum
-
-def solve_ode_deep_neural_network(x, num_neurons, num_iter, lmb):
-    # num_hidden_neurons is now a list of number of neurons within each hidden layer
-
-    # Find the number of hidden layers:
-    N_hidden = np.size(num_neurons)
-    
-    ## Set up initial weigths and biases 
-    
-    # Initialize the list of parameters:
-    P = [None]*(N_hidden + 1) # + 1 to include the output layer
-
-    P[0] = npr.randn(num_neurons[0], 2 ) 
-    for l in range(1,N_hidden):
-        P[l] = npr.randn(num_neurons[l], num_neurons[l-1] + 1) # +1 to include bias 
-    
-    # For the output layer
-    P[-1] = npr.randn(1, num_neurons[-1] + 1 ) # +1 since bias is included
-
-    print('Initial cost: %g'%cost_function_deep(P, x))
-    
-    ## Start finding the optimal weigths using gradient descent
-    
-    # Find the Python function that represents the gradient of the cost function
-    # w.r.t the 0-th input argument -- that is the weights and biases in the hidden and output layer
-    cost_function_deep_grad = grad(cost_function_deep,0)
-    
-    # Let the update be done num_iter times
-    for i in range(num_iter):
-        # Evaluate the gradient at the current weights and biases in P. 
-        # The cost_grad consist now of N_hidden + 1 arrays; the gradient w.r.t the weights and biases
-        # in the hidden layers and output layers evaluated at x.
-        cost_deep_grad =  cost_function_deep_grad(P, x)
-        
-        for l in range(N_hidden+1):
-            P[l] = P[l] - lmb * cost_deep_grad[l]
-
-    print('Final cost: %g'%cost_function_deep(P, x))
-    
-    return P
-
    -
    - - -
    -

    Solving the ODE

    - -

    -Finally, having set up the networks we are ready to use them to solve the ODE problem. -We add the analytical solution - -

    - - -

    def g_analytic(x, gamma = 2, g0 = 10):
-    return g0*np.exp(-gamma*x)
-
    -
    - - -
    -

    Using neural network

    - -

    -The code below solves the ODE using a neural network. The number of -values for the input \( \vec x \) is 10, number of hidden neurons in the -hidden layer being 10 and th step size used in gradien descent -\( \lambda = 0.001 \). The program updates the weights and biases in the -network for a given number of iterations. Finally, it plots the results from using the -neural network along with the analytical solution. - -

    - - -

    npr.seed(15)
-
-## Decide the vales of arguments to the function to solve
-N = 10
-x = np.linspace(0, 1, N)
-
-## Set up the initial parameters
-num_hidden_neurons = 10
-num_iter = 10000
-lmb = 0.001
-
-P = solve_ode_neural_network(x, num_hidden_neurons, num_iter, lmb)
-
-res = g_trial(x,P) 
-res_analytical = g_analytic(x)
-
-print('Max absolute difference: %g'%np.max(np.abs(res - res_analytical)))
-
-plt.figure(figsize=(10,10))
-
-plt.title('Performance of neural network solving an ODE compared to the analytical solution')
-plt.plot(x, res_analytical)
-plt.plot(x, res[0,:])
-plt.legend(['analytical','nn'])
-plt.xlabel('x')
-plt.ylabel('g(x)')
-plt.show()
-
    -
    - - -
    -

    Using a deep neural network

    - -

    - - -

    npr.seed(15)
-
-## Decide the vales of arguments to the function to solve
-N = 10
-x = np.linspace(0, 1, N)
-
-## Set up the initial parameters
-num_hidden_neurons = np.array([10,10])
-num_iter = 10000
-lmb = 0.001
-
-P = solve_ode_deep_neural_network(x, num_hidden_neurons, num_iter, lmb)
-
-res = g_trial_deep(x,P) 
-res_analytical = g_analytic(x)
-
-plt.figure(figsize=(10,10))
-
-plt.title('Performance of a deep neural network solving an ODE compared to the analytical solution')
-plt.plot(x, res_analytical)
-plt.plot(x, res[0,:])
-plt.legend(['analytical','dnn'])
-plt.ylabel('g(x)')
-plt.show()
-
    -
    - - -
    -

    Wrapping it up

    - -

    -By rewriting the ODE as a minimization problem, it was possible to -solve equation using either a neural network (one hidden layer) or a -deep neural network (more than one hidden layers). How well the -network performed is measured by a specified cost function, which is -the function the network tries to minimize. Using a trial solution -which satisfies the additional condition and being defined by using -the output from the network in some way, the minimization problem -could be explicitly defined for out network to solve. The proposed -solution from the network is then the trial solution with parameters, -that is weights and biases within each layer in the network, such that -the solution minimizes the cost function. -

    - -
    diff --git a/doc/pub/NeuralNet/html/NeuralNet-solarized.html b/doc/pub/NeuralNet/html/NeuralNet-solarized.html index c2cc3ebea..4a0d1d535 100644 --- a/doc/pub/NeuralNet/html/NeuralNet-solarized.html +++ b/doc/pub/NeuralNet/html/NeuralNet-solarized.html @@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source - + -Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks +Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning @@ -161,65 +161,7 @@ div { text-align: justify; text-justify: inter-word; } ('Limitations of supervised learning with deep networks', 2, None, - '___sec65'), - ('Convolutional Neural Networks (recognizing images)', - 2, - None, - '___sec66'), - ('Regular NNs don’t scale well to full images', - 2, - None, - '___sec67'), - ('3D volumes of neurons', 2, None, '___sec68'), - ('Layers used to build CNNs', 2, None, '___sec69'), - ('Transforming images', 2, None, '___sec70'), - ('CNNs in brief', 2, None, '___sec71'), - ('CNNs in more detail, building convolutional neural networks in ' - 'Tensorflow and Keras', - 2, - None, - '___sec72'), - ('Setting it up', 2, None, '___sec73'), - ('The MNIST dataset again', 2, None, '___sec74'), - ('Strong correlations', 2, None, '___sec75'), - ('Layers of a CNN', 2, None, '___sec76'), - ('Systematic reduction', 2, None, '___sec77'), - ('Prerequisites: Collect and pre-process data', - 2, - None, - '___sec78'), - ('Importing Keras and Tensorflow', 2, None, '___sec79'), - ('Using TensorFlow backend', 2, None, '___sec80'), - ('Train the model', 2, None, '___sec81'), - ('Visualizing the results', 2, None, '___sec82'), - ('Running with Keras', 2, None, '___sec83'), - ('Final part', 2, None, '___sec84'), - ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86'), - ('Applications: solving ordinary differential equations with ' - 'Neural Networks', - 2, - None, - '___sec87'), - ('Trial solution', 2, None, '___sec88'), - ('More details', 2, None, '___sec89'), - ('Reformulating the problem', 2, None, '___sec90'), - ('Estimating errors', 2, None, '___sec91'), - ('Creating a simple Deep Neural Net', 2, None, '___sec92'), - ('Setting up the code, feed forward part', 2, None, '___sec93'), - ('Backpropagation', 2, None, '___sec94'), - ('Gradient Descent', 2, None, '___sec95'), - ('More on GD and cost function', 2, None, '___sec96'), - ('An implementation of a Deep Neural Network', - 2, - None, - '___sec97'), - ('The final parts of the code', 2, None, '___sec98'), - ('And adding Back propagation', 2, None, '___sec99'), - ('Solving the ODE', 2, None, '___sec100'), - ('Using neural network', 2, None, '___sec101'), - ('Using a deep neural network', 2, None, '___sec102'), - ('Wrapping it up', 2, None, '___sec103')]} + '___sec65')]} end of tocinfo --> @@ -245,7 +187,7 @@ MathJax.Hub.Config({ -

    Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks

    +

    Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning

    @@ -2983,1317 +2925,6 @@ Here we list some of the important limitations of supervised neural network base Some of these remarks are particular to DNNs, others are shared by all supervised learning methods. This motivates the use of unsupervised methods which in part circumvent these problems.

    -









    - -

    Convolutional Neural Networks (recognizing images)

    - -

    -Convolutional Neural Networks (CNN) are very similar to ordinary Neural Networks. - -

    -They are made up of neurons that have learnable weights and -biases. Each neuron receives some inputs, performs a dot product and -optionally follows it with a non-linearity. The whole network still -expresses a single differentiable score function: from the raw image -pixels on one end to class scores at the other. And they still have a -loss function (for example Softmax) on the last (fully-connected) layer -and all the tips/tricks we developed for learning regular Neural -Networks still apply (back propagation, gradient descent etc etc). - -

    -What is the difference? CNN architectures make the explicit assumption that -the inputs are images, which allows us to encode certain properties -into the architecture. These then make the forward function more -efficient to implement and vastly reduce the amount of parameters in -the network. - -

    -Here we provide only a superficial overview, for the more interested, we recommend highly the course -IN5400 – Machine Learning for Image Analysis -and the slides of CS231. - -

    -









    - -

    Regular NNs don’t scale well to full images

    - -

    -As an example, consider -an image of size \( 32\times 32\times 3 \) (32 wide, 32 high, 3 color channels), so a -single fully-connected neuron in a first hidden layer of a regular -Neural Network would have \( 32\times 32\times 3 = 3072 \) weights. This amount still -seems manageable, but clearly this fully-connected structure does not -scale to larger images. For example, an image of more respectable -size, say \( 200\times 200\times 3 \), would lead to neurons that have -\( 200\times 200\times 3 = 120,000 \) weights. - -

    -We could have -several such neurons, and the parameters would add up quickly! Clearly, -this full connectivity is wasteful and the huge number of parameters -would quickly lead to possible overfitting. - -

    -

    -
    -

    Figure 1: A regular 3-layer Neural Network.

    -

    -
    - -

    -









    - -

    3D volumes of neurons

    - -

    -Convolutional Neural Networks take advantage of the fact that the -input consists of images and they constrain the architecture in a more -sensible way. - -

    -In particular, unlike a regular Neural Network, the -layers of a CNN have neurons arranged in 3 dimensions: width, -height, depth. (Note that the word depth here refers to the third -dimension of an activation volume, not to the depth of a full Neural -Network, which can refer to the total number of layers in a network.) - -

    -To understand it better, the above example of an image -with an input volume of -activations has dimensions \( 32\times 32\times 3 \) (width, height, -depth respectively). - -

    -The neurons in a layer will -only be connected to a small region of the layer before it, instead of -all of the neurons in a fully-connected manner. Moreover, the final -output layer could for this specific image have dimensions \( 1\times 1 \times 10 \), -because by the -end of the CNN architecture we will reduce the full image into a -single vector of class scores, arranged along the depth -dimension. - -

    -

    -
    -

    Figure 2: A CNN arranges its neurons in three dimensions (width, height, depth), as visualized in one of the layers. Every layer of a CNN transforms the 3D input volume to a 3D output volume of neuron activations. In this example, the red input layer holds the image, so its width and height would be the dimensions of the image, and the depth would be 3 (Red, Green, Blue channels).

    -

    -
    - -

    - - -

    Layers used to build CNNs

    - -

    -A simple CNN is a sequence of layers, and every layer of a CNN -transforms one volume of activations to another through a -differentiable function. We use three main types of layers to build -CNN architectures: Convolutional Layer, Pooling Layer, and -Fully-Connected Layer (exactly as seen in regular Neural Networks). We -will stack these layers to form a full CNN architecture. - -

    -A simple CNN for image classification could have the architecture: - -

    - -









    - -

    Transforming images

    - -

    -CNNs transform the original image layer by layer from the original -pixel values to the final class scores. - -

    -Observe that some layers contain -parameters and other don’t. In particular, the CNN layers perform -transformations that are a function of not only the activations in the -input volume, but also of the parameters (the weights and biases of -the neurons). On the other hand, the RELU/POOL layers will implement a -fixed function. The parameters in the CONV/FC layers will be trained -with gradient descent so that the class scores that the CNN computes -are consistent with the labels in the training set for each image. - -

    -









    - -

    CNNs in brief

    - -

    -In summary: - -

    - -For more material on convolutional networks, we strongly recommend -the course -IN5400 – Machine Learning for Image Analysis -and the slides of CS231 which is taught at Stanford University (consistently ranked as one of the top computer science programs in the world). Michael Nielsen's book is a must read, in particular chapter 6 which deals with CNNs. - -

    -









    - -

    CNNs in more detail, building convolutional neural networks in Tensorflow and Keras

    - -

    -As discussed above, CNNs are neural networks built from the assumption that the inputs -to the network are 2D images. This is important because the number of features or pixels in images -grows very fast with the image size, and an enormous number of weights and biases are needed in order to build an accurate network. - -

    -As before, we still have our input, a hidden layer and an output. What's novel about convolutional networks -are the convolutional and pooling layers stacked in pairs between the input and the hidden layer. -In addition, the data is no longer represented as a 2D feature matrix, instead each input is a number of 2D -matrices, typically 1 for each color dimension (Red, Green, Blue). - -

    -









    - -

    Setting it up

    - -

    -It means that to represent the entire -dataset of images, we require a 4D matrix or tensor. This tensor has the dimensions: -$$ -(n_{inputs},\, n_{pixels, width},\, n_{pixels, height},\, depth) . -$$ - -

    -









    - -

    The MNIST dataset again

    - -

    -The MNIST dataset consists of grayscale images with a pixel size of -\( 28\times 28 \), meaning we require \( 28 \times 28 = 724 \) weights to each -neuron in the first hidden layer. - -

    -If we were to analyze images of size \( 128\times 128 \) we would require -\( 128 \times 128 = 16384 \) weights to each neuron. Even worse if we were -dealing with color images, as most images are, we have an image matrix -of size \( 128\times 128 \) for each color dimension (Red, Green, Blue), -meaning 3 times the number of weights \( = 49152 \) are required for every -single neuron in the first hidden layer. - -

    -









    - -

    Strong correlations

    -Images typically have strong local correlations, meaning that a small -part of the image varies little from its neighboring regions. If for -example we have an image of a blue car, we can roughly assume that a -small blue part of the image is surrounded by other blue regions. - -

    -Therefore, instead of connecting every single pixel to a neuron in the -first hidden layer, as we have previously done with deep neural -networks, we can instead connect each neuron to a small part of the -image (in all 3 RGB depth dimensions). The size of each small area is -fixed, and known as a receptive. - -

    - - -

    Layers of a CNN

    -The layers of a convolutional neural network arrange neurons in 3D: width, height and depth. -The input image is typically a square matrix of depth 3. - -

    -A convolution is performed on the image which outputs -a 3D volume of neurons. The weights to the input are arranged in a number of 2D matrices, known as filters. - -

    -Each filter slides along the input image, taking the dot product -between each small part of the image and the filter, in all depth -dimensions. This is then passed through a non-linear function, -typically the Rectified Linear (ReLu) function, which serves as the -activation of the neurons in the first convolutional layer. This is -further passed through a pooling layer, which reduces the size of the -convolutional layer, e.g. by taking the maximum or average across some -small regions, and this serves as input to the next convolutional -layer. - -

    -









    - -

    Systematic reduction

    - -

    -By systematically reducing the size of the input volume, through -convolution and pooling, the network should create representations of -small parts of the input, and then from them assemble representations -of larger areas. The final pooling layer is flattened to serve as -input to a hidden layer, such that each neuron in the final pooling -layer is connected to every single neuron in the hidden layer. This -then serves as input to the output layer, e.g. a softmax output for -classification. - -

    -









    - -

    Prerequisites: Collect and pre-process data

    -

    - - -

    # import necessary packages
-import numpy as np
-import matplotlib.pyplot as plt
-from sklearn import datasets
-
-
-# ensure the same random numbers appear every time
-np.random.seed(0)
-
-# display images in notebook
-%matplotlib inline
-plt.rcParams['figure.figsize'] = (12,12)
-
-
-# download MNIST dataset
-digits = datasets.load_digits()
-
-# define inputs and labels
-inputs = digits.images
-labels = digits.target
-
-# RGB images have a depth of 3
-# our images are grayscale so they should have a depth of 1
-inputs = inputs[:,:,:,np.newaxis]
-
-print("inputs = (n_inputs, pixel_width, pixel_height, depth) = " + str(inputs.shape))
-print("labels = (n_inputs) = " + str(labels.shape))
-
-
-# choose some random images to display
-n_inputs = len(inputs)
-indices = np.arange(n_inputs)
-random_indices = np.random.choice(indices, size=5)
-
-for i, image in enumerate(digits.images[random_indices]):
-    plt.subplot(1, 5, i+1)
-    plt.axis('off')
-    plt.imshow(image, cmap=plt.cm.gray_r, interpolation='nearest')
-    plt.title("Label: %d" % digits.target[random_indices[i]])
-plt.show()
-
    -

    -









    - -

    Importing Keras and Tensorflow

    -

    - - -

    from keras.utils import to_categorical
-from sklearn.model_selection import train_test_split
-
-# representation of labels
-labels = to_categorical(labels)
-
-# split into train and test data
-# one-liner from scikit-learn library
-train_size = 0.8
-test_size = 1 - train_size
-X_train, X_test, Y_train, Y_test = train_test_split(inputs, labels, train_size=train_size,
-                                                    test_size=test_size)
-
    -

    -









    - -

    Using TensorFlow backend

    - -

    -We need to define model and architecture and choose cost function and optmizer. -

    - - -

    import tensorflow as tf
-
-class ConvolutionalNeuralNetworkTensorflow:
-    def __init__(
-            self,
-            X_train,
-            Y_train,
-            X_test,
-            Y_test,
-            n_filters=10,
-            n_neurons_connected=50,
-            n_categories=10,
-            receptive_field=3,
-            stride=1,
-            padding=1,
-            epochs=10,
-            batch_size=100,
-            eta=0.1,
-            lmbd=0.0):
-        
-        self.global_step = tf.Variable(0, dtype=tf.int32, trainable=False, name='global_step')
-        
-        self.X_train = X_train
-        self.Y_train = Y_train
-        self.X_test = X_test
-        self.Y_test = Y_test
-        
-        self.n_inputs, self.input_width, self.input_height, self.depth = X_train.shape
-        
-        self.n_filters = n_filters
-        self.n_downsampled = int(self.input_width*self.input_height*n_filters / 4)
-        self.n_neurons_connected = n_neurons_connected
-        self.n_categories = n_categories
-        
-        self.receptive_field = receptive_field
-        self.stride = stride
-        self.strides = [stride, stride, stride, stride]
-        self.padding = padding
-        
-        self.epochs = epochs
-        self.batch_size = batch_size
-        self.iterations = self.n_inputs // self.batch_size
-        self.eta = eta
-        self.lmbd = lmbd
-        
-        self.create_placeholders()
-        self.create_CNN()
-        self.create_loss()
-        self.create_optimiser()
-        self.create_accuracy()
-    
-    def create_placeholders(self):
-        with tf.name_scope('data'):
-            self.X = tf.placeholder(tf.float32, shape=(None, self.input_width, self.input_height, self.depth), name='X_data')
-            self.Y = tf.placeholder(tf.float32, shape=(None, self.n_categories), name='Y_data')
-    
-    def create_CNN(self):
-        with tf.name_scope('CNN'):
-            
-            # Convolutional layer
-            self.W_conv = self.weight_variable([self.receptive_field, self.receptive_field, self.depth, self.n_filters], name='conv', dtype=tf.float32)
-            b_conv = self.weight_variable([self.n_filters], name='conv', dtype=tf.float32)
-            z_conv = tf.nn.conv2d(self.X, self.W_conv, self.strides, padding='SAME', name='conv') + b_conv
-            a_conv = tf.nn.relu(z_conv)
-            
-            # 2x2 max pooling
-            a_pool = tf.nn.max_pool(a_conv, [1, 2, 2, 1], [1, 2, 2, 1], padding='SAME', name='pool')
-            
-            # Fully connected layer
-            a_pool_flat = tf.reshape(a_pool, [-1, self.n_downsampled])
-            self.W_fc = self.weight_variable([self.n_downsampled, self.n_neurons_connected], name='fc', dtype=tf.float32)
-            b_fc = self.bias_variable([self.n_neurons_connected], name='fc', dtype=tf.float32)
-            a_fc = tf.nn.relu(tf.matmul(a_pool_flat, self.W_fc) + b_fc)
-            
-            # Output layer
-            self.W_out = self.weight_variable([self.n_neurons_connected, self.n_categories], name='out', dtype=tf.float32)
-            b_out = self.bias_variable([self.n_categories], name='out', dtype=tf.float32)
-            self.z_out = tf.matmul(a_fc, self.W_out) + b_out
-    
-    def create_loss(self):
-        with tf.name_scope('loss'):
-            softmax_loss = tf.reduce_mean(tf.nn.softmax_cross_entropy_with_logits_v2(labels=self.Y, logits=self.z_out))
-            
-            regularizer_loss_conv = tf.nn.l2_loss(self.W_conv)
-            regularizer_loss_fc = tf.nn.l2_loss(self.W_fc)
-            regularizer_loss_out = tf.nn.l2_loss(self.W_out)
-            regularizer_loss = self.lmbd*(regularizer_loss_conv + regularizer_loss_fc + regularizer_loss_out)
-            
-            self.loss = softmax_loss + regularizer_loss
-
-    def create_accuracy(self):
-        with tf.name_scope('accuracy'):
-            probabilities = tf.nn.softmax(self.z_out)
-            predictions = tf.argmax(probabilities, 1)
-            labels = tf.argmax(self.Y, 1)
-            
-            correct_predictions = tf.equal(predictions, labels)
-            correct_predictions = tf.cast(correct_predictions, tf.float32)
-            self.accuracy = tf.reduce_mean(correct_predictions)
-    
-    def create_optimiser(self):
-        with tf.name_scope('optimizer'):
-            self.optimizer = tf.train.GradientDescentOptimizer(learning_rate=self.eta).minimize(self.loss, global_step=self.global_step)
-            
-    def weight_variable(self, shape, name='', dtype=tf.float32):
-        initial = tf.truncated_normal(shape, stddev=0.1)
-        return tf.Variable(initial, name=name, dtype=dtype)
-    
-    def bias_variable(self, shape, name='', dtype=tf.float32):
-        initial = tf.constant(0.1, shape=shape)
-        return tf.Variable(initial, name=name, dtype=dtype)
-
-    def fit(self):
-        data_indices = np.arange(self.n_inputs)
-
-        with tf.Session() as sess:
-            sess.run(tf.global_variables_initializer())
-            for i in range(self.epochs):
-                for j in range(self.iterations):
-                    chosen_datapoints = np.random.choice(data_indices, size=self.batch_size, replace=False)
-                    batch_X, batch_Y = self.X_train[chosen_datapoints], self.Y_train[chosen_datapoints]
-            
-                    sess.run([CNN.loss, CNN.optimizer],
-                        feed_dict={CNN.X: batch_X,
-                                   CNN.Y: batch_Y})
-                    accuracy = sess.run(CNN.accuracy,
-                        feed_dict={CNN.X: batch_X,
-                                   CNN.Y: batch_Y})
-                    step = sess.run(CNN.global_step)
-    
-            self.train_loss, self.train_accuracy = sess.run([CNN.loss, CNN.accuracy],
-                feed_dict={CNN.X: self.X_train,
-                           CNN.Y: self.Y_train})
-        
-            self.test_loss, self.test_accuracy = sess.run([CNN.loss, CNN.accuracy],
-                feed_dict={CNN.X: self.X_test,
-                           CNN.Y: self.Y_test})
-
    -

    -









    - -

    Train the model

    - -

    -We need now to train the model, evaluate it and test its performance on test data, and eventually include hyperparameters. -

    - - -

    epochs = 100
-batch_size = 100
-n_filters = 10
-n_neurons_connected = 50
-n_categories = 10
-
-eta_vals = np.logspace(-5, 1, 7)
-lmbd_vals = np.logspace(-5, 1, 7)
-CNN_tf = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)
-        
-for i, eta in enumerate(eta_vals):
-    for j, lmbd in enumerate(lmbd_vals):
-        CNN = ConvolutionalNeuralNetworkTensorflow(X_train, Y_train, X_test, Y_test,
-                                      n_filters=n_filters, n_neurons_connected=n_neurons_connected,
-                                      n_categories=n_categories, epochs=epochs, batch_size=batch_size,
-                                      eta=eta, lmbd=lmbd)
-        CNN.fit()
-        
-        print("Learning rate = ", eta)
-        print("Lambda = ", lmbd)
-        print("Test accuracy: %.3f" % CNN.test_accuracy)
-        print()
-            
-        CNN_tf[i][j] = CNN
-
    -

    -









    - -

    Visualizing the results

    - -

    - - -

    # visual representation of grid search
-# uses seaborn heatmap, could probably do this in matplotlib
-import seaborn as sns
-
-sns.set()
-
-train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
-test_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
-
-for i in range(len(eta_vals)):
-    for j in range(len(lmbd_vals)):
-        CNN = CNN_tf[i][j]
-
-        train_accuracy[i][j] = CNN.train_accuracy
-        test_accuracy[i][j] = CNN.test_accuracy
-
-        
-fig, ax = plt.subplots(figsize = (10, 10))
-sns.heatmap(train_accuracy, annot=True, ax=ax, cmap="viridis")
-ax.set_title("Training Accuracy")
-ax.set_ylabel("$\eta$")
-ax.set_xlabel("$\lambda$")
-plt.show()
-
-fig, ax = plt.subplots(figsize = (10, 10))
-sns.heatmap(test_accuracy, annot=True, ax=ax, cmap="viridis")
-ax.set_title("Test Accuracy")
-ax.set_ylabel("$\eta$")
-ax.set_xlabel("$\lambda$")
-plt.show()
-
    -

    - - -

    Running with Keras

    - -

    - - -

    from keras.models import Sequential
-from keras.layers.convolutional import Conv2D
-from keras.layers.convolutional import MaxPooling2D
-from keras.layers import Flatten
-from keras.layers import Dense
-from keras.regularizers import l2
-from keras.optimizers import SGD
-
-def create_convolutional_neural_network_keras(input_shape, receptive_field,
-                                              n_filters, n_neurons_connected, n_categories,
-                                              eta, lmbd):
-    model = Sequential()
-    model.add(Conv2D(n_filters, (receptive_field, receptive_field), input_shape=input_shape, padding='same',
-              activation='relu', kernel_regularizer=l2(lmbd)))
-    model.add(MaxPooling2D(pool_size=(2, 2)))
-    model.add(Flatten())
-    model.add(Dense(n_neurons_connected, activation='relu', kernel_regularizer=l2(lmbd)))
-    model.add(Dense(n_categories, activation='softmax', kernel_regularizer=l2(lmbd)))
-    
-    sgd = SGD(lr=eta)
-    model.compile(loss='categorical_crossentropy', optimizer=sgd, metrics=['accuracy'])
-    
-    return model
-
-epochs = 100
-batch_size = 100
-input_shape = X_train.shape[1:4]
-receptive_field = 3
-n_filters = 10
-n_neurons_connected = 50
-n_categories = 10
-
-eta_vals = np.logspace(-5, 1, 7)
-lmbd_vals = np.logspace(-5, 1, 7)
-
    -

    -









    - -

    Final part

    - -

    - - -

    CNN_keras = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)
-        
-for i, eta in enumerate(eta_vals):
-    for j, lmbd in enumerate(lmbd_vals):
-        CNN = create_convolutional_neural_network_keras(input_shape, receptive_field,
-                                              n_filters, n_neurons_connected, n_categories,
-                                              eta, lmbd)
-        CNN.fit(X_train, Y_train, epochs=epochs, batch_size=batch_size, verbose=0)
-        scores = CNN.evaluate(X_test, Y_test)
-        
-        CNN_keras[i][j] = CNN
-        
-        print("Learning rate = ", eta)
-        print("Lambda = ", lmbd)
-        print("Test accuracy: %.3f" % scores[1])
-        print()
-
    -

    -









    - -

    Final visualization

    - -

    - - -

    # visual representation of grid search
-# uses seaborn heatmap, could probably do this in matplotlib
-import seaborn as sns
-
-sns.set()
-
-train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
-test_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
-
-for i in range(len(eta_vals)):
-    for j in range(len(lmbd_vals)):
-        CNN = CNN_keras[i][j]
-
-        train_accuracy[i][j] = CNN.evaluate(X_train, Y_train)[1]
-        test_accuracy[i][j] = CNN.evaluate(X_test, Y_test)[1]
-
-        
-fig, ax = plt.subplots(figsize = (10, 10))
-sns.heatmap(train_accuracy, annot=True, ax=ax, cmap="viridis")
-ax.set_title("Training Accuracy")
-ax.set_ylabel("$\eta$")
-ax.set_xlabel("$\lambda$")
-plt.show()
-
-fig, ax = plt.subplots(figsize = (10, 10))
-sns.heatmap(test_accuracy, annot=True, ax=ax, cmap="viridis")
-ax.set_title("Test Accuracy")
-ax.set_ylabel("$\eta$")
-ax.set_xlabel("$\lambda$")
-plt.show()
-
    -

    -









    - -

    Fun links

    - -
      -
    1. Self-Driving cars using a convolutional neural network
      2. -
    2. Abstract art using convolutional neural networks
      3. -
    - -









    - -

    Applications: solving ordinary differential equations with Neural Networks

    - -

    -We end our discussion on neural networks with a discussion on how to solve differential equations. Here we focus -first on the classical exponential decay in one dimension. Thereafter we switch to the Poisson equation in one dimension. - -

    -The aim is to see if we can use a neural network to solve - -$$ -\begin{equation} -\label{eq:ode} -g'(x) = -\gamma g(x) -\end{equation} -$$ - -

    -where \( g(0) = g_0 \) with \( \gamma \) and \( g_0 \) being some chosen -values. This equation is an ordinary differential equation since the -function we have to solve for, \( g(x) \), is of one variable. - -

    -Here we set \( \gamma = 2 \) and \( g_0 = 10 \) but feel free to change -them and see how the neural network performs. - -

    -









    - -

    Trial solution

    - -

    -To begin with, a trial solution \( g_t(t) \) must be chosen. A general -trial solution for ordinary differential equations could be - -$$ -g_t(x, P) = h_1(x) + h_2(x, N(x, P)), -$$ - -

    -with \( h_1(x) \) ensuring that \( g_t(x) \) satisfies some conditions and -\( h_2(x,N(x, P)) \) an expression involving \( x \) and the output from the -neural network \( N(x,P) \) with \( P \) being the collection of the weights -and biases for each layer. - -

    -It is assumed that there are no weights and -bias at the input layer, so \( P = \{ P_{\mathrm{hidden}}, -P_{\mathrm{output}} \} \). If there are \( N_{\mathrm{hidden} } \) neurons in -the hidden layer, then \( P_{\mathrm{hidden}} \) is an \( N_{\mathrm{hidden} } -\times 2 \) matrix. - -

    -The first column in \( P_{\mathrm{hidden} } \) represents -the bias for each neuron in the hidden layer and the second column -represents the weigths for each neuron. If there are \( N_{\mathrm{output} -} \) neurons in the output layer, then \( P_{\mathrm{output}} \) is a -\( N_{\mathrm{output} } \times (1 + N_{\mathrm{hidden} }) \) matrix. Its first -column represents the bias of each neuron and the remaining columns -represents the weights to each neuron. - -

    -









    - -

    More details

    - -

    -We have \( g(0) = g_0 \). The trial solution must fulfill this -condition to be a proper solution of \eqref{eq:ode}. - -

    -A possible way to -ensure that \( g_t(0, P) = g_0 \), is to let \( F(N(x,P)) = x\cdot N(x,P) \) -and \( A(x) = g_0 \). This gives the following trial solution: - -$$ -\begin{equation} -g_t(x, P) = g_0 + x \cdot N(x, P). -\label{_auto11} -\end{equation} -$$ - -

    -









    - -

    Reformulating the problem

    - -

    -Often, the role of a neural network is to minimize its parameters with -respect to some given error criteria. This criteria, the cost or loss -function, is a measure of how much error the output of the network has -compared to some given known answers. A reformulation of -\eqref{eq:ode} must therefore be done, such that it describes the -problem a neural network can solve. - -

    -The neural network must find the set of weigths and biases \( P \) such -that the trial solution in satisfies -\eqref{eq:ode}. The trial solution has been chosen such that it -already solves the condition \( g(0) = g_0 \). What remains, is to find -\( P \) such that - -$$ -\begin{equation} -g_t'(x, P) = - \gamma g_t(x, P) -\label{_auto12} -\end{equation} -$$ - -

    -is fulfilled as best as possible. - -

    -









    - -

    Estimating errors

    - -

    -Having two sides of an equation as equal as -possible, means that the absolute or squared difference between the -sides must be as close to zero as small. In this case, the difference -squared is an appropiate measurement of how errorneous the trial -solution is with respect to \( P \) of the neural network. Therefore, the -problem our network must solve, is - -$$ -\min_{P}\Big\{ \big(g_t'(x, P) - ( -\gamma g_t(x, P) \big)^2 \Big\} -$$ - -

    -or, in terms of weights and biases for each layer: - -$$ -\min_{P_{\mathrm{hidden} }, \ P_{\mathrm{output} }}\Big\{ \big(g_t'(x, \{ P_{\mathrm{hidden} }, P_{\mathrm{output} }\}) - ( -\gamma g_t(x, \{ P_{\mathrm{hidden} }, P_{\mathrm{output} }\}) \big)^2 \Big\} -$$ - -

    -for an input value \( x \). -If the neural network evaluates \( g_t(x, P) \) at more avalues for \( x \), say \( N \) values \( x_i \) for \( i = 1, \dots, N \), then the total error to minimize is - -$$ -\begin{equation} -\label{eq:min} -\min_{P}\Big\{\sum_i \big(g_t'(x_i, P) - ( -\gamma g_t(x_i, P) \big)^2 \Big\} -\end{equation} -$$ - -

    -Letting \( c(x, P) = \sum_i \big(g_t'(x_i, P) - ( -\gamma g_t(x_i, P) -\big)^2 \) denote the cost function, the minimization problem of which -our network must solve, is - -$$ -\min_{P} c(x, P) -$$ - -

    -or in terms of \( P_{\mathrm{hidden} } \) and \( P_{\mathrm{output} } \) - -$$ -\min_{P_{\mathrm{hidden} }, \ P_{\mathrm{output} }} c(x, \{P_{\mathrm{hidden} }, P_{\mathrm{output} }\}) -$$ - -

    -









    - -

    Creating a simple Deep Neural Net

    - -

    -The next step is to decide how the neural net \( N(x, P) \) -should be. In this case, the neural network is made -from scratch to understand better how a neural network works, gain -more control over its architecture, and see how Autograd can be used -to simplify the implementation. - -

    -Since a deep neural network (DNN) is a neural network with more than -one hidden layer, we can first look on how to implement a neural -network. Having an implementation of a neural network at hand, an -extension of it into a deep neural network would (hopefully) be -painless. - -

    -For simplicity, we assume that the input is an array -\( \hat{x}= (x_1, \dots, x_N) \) with \( N \) elements. It is at these points the neural -network should find \( P \) such that it fulfills \eqref{eq:min}. -All the ingredients discussed earlier, from the activation function, hidden layers and their weights, biases etc -are included below. - -

    -









    - -

    Setting up the code, feed forward part

    -

    - - -

    # Note that we use the  numpy wrapper for Autograd (see the gradient descent  slides)
-import autograd.numpy as np
-from autograd import grad, elementwise_grad
-import autograd.numpy.random as npr
-from matplotlib import pyplot as plt
-
-def sigmoid(z):
-    return 1/(1 + np.exp(-z))
-
-def neural_network(params, x):
-    
-    # Find the weights (including and biases) for the hidden and output layer.
-    # Assume that params is a list of parameters for each layer. 
-    # The biases are the first element for each array in params, 
-    # and the weights are the remaning elements in each array in params.   
-    
-    w_hidden = params[0]
-    w_output = params[1]
-
-    # Assumes input x being an one-dimensional array
-    num_values = np.size(x)
-    x = x.reshape(-1, num_values)
-    
-    # Assume that the input layer does nothing to the input x
-    x_input = x
-
-    ## Hidden layer:
-    
-    # Add a row of ones to include bias
-    x_input = np.concatenate((np.ones((1,num_values)), x_input ), axis = 0)
-    
-    z_hidden = np.matmul(w_hidden, x_input)
-    x_hidden = sigmoid(z_hidden)
-
-    ## Output layer:
-    
-    # Include bias:
-    x_hidden = np.concatenate((np.ones((1,num_values)), x_hidden ), axis = 0)
-
-    z_output = np.matmul(w_output, x_hidden)
-    x_output = z_output
-
-    return x_output
-
    -

    -









    - -

    Backpropagation

    - -

    -Now that the feedforward can be done, the next step is to decide how the -parameters should change such that they minimize the cost function. - -

    -Recall that the chosen cost function for this problem is - -$$ -c(x, P) = \sum_i \big(g_t'(x_i, P) - ( -\gamma g_t(x_i, P) \big)^2 -$$ - -

    -In order to minimize it, an optimization method must be chosen. - -

    -Here, gradient descent with a constant step size has been chosen. - -

    -Before looking at the gradient descent method, let us set up the cost -function along with the right ride of the ODE and trial solution. - -

    - - -

    # The trial solution using the deep neural network:
-def g_trial(x,params, g0 = 10):
-    return g0 + x*neural_network(params,x)
-
-# The right side of the ODE:
-def g(x, g_trial, gamma = 2):
-    return -gamma*g_trial
-
-# The cost function:
-def cost_function(P, x):
-    
-    # Evaluate the trial function with the current parameters P
-    g_t = g_trial(x,P)
-    
-    # Find the derivative w.r.t x of the neural network
-    d_net_out = elementwise_grad(neural_network,1)(P,x) 
-    
-    # Find the derivative w.r.t x of the trial function
-    d_g_t = elementwise_grad(g_trial,0)(x,P)  
-    
-    # The right side of the ODE 
-    func = g(x, g_t)
-
-    err_sqr = (d_g_t - func)**2
-    cost_sum = np.sum(err_sqr)
-    
-    return cost_sum
-
    -

    -









    - -

    Gradient Descent

    - -

    -The idea of the gradient descent algorithm is to update parameters in -direction where the cost function decreases goes to a minimum. - -

    -In general, the update of some parameters \( \hat{\omega} \) given a cost -function defined by some weights \( \hat{\omega} \), \( c(x, \hat{\omega}) \), -goes as follows: - -$$ -\hat{\omega}_{\mathrm{new} } = \hat{\omega} - \lambda \nabla_{\hat{\omega}} c(x, \hat{\omega}), -$$ - -

    -for a number of iterations or until \( \big|\big| \hat{\omega}_{\mathrm{new} } - \hat{\omega} \big|\big| \) -is smaller than some -given tolerance. - -

    -The value of \( \lambda \) decides how large steps the algorithm must take -in the direction of $ \nabla_{\hat{\omega}} c(x, \hat{\omega})$. The -notatation \( \nabla_{\hat{\omega}} \) denotes the gradient with respect to -the elements in \( \hat{\omega} \). - -

    -









    - -

    More on GD and cost function

    - -

    -In our case, we have to minimize the cost function \( c(x, P) \) with -respect to the two sets of weights and bisases, that is for the hidden -layer \( P_{\mathrm{hidden} } \) and for the ouput layer \( P_{\mathrm{output} -} \) . - -

    -This means that \( P_{\mathrm{hidden} } \) and \( P_{\mathrm{output} } \) is -updated by - -$$ -\begin{align} -P_{\mathrm{hidden},\mathrm{new}} &= P_{\mathrm{hidden}} - \lambda \nabla_{P_{\mathrm{hidden}}} c(x, P) -\label{_auto13}\\ -P_{\mathrm{output},\mathrm{new}} &= P_{\mathrm{output}} - \lambda \nabla_{P_{\mathrm{output}}} c(x, P) -\label{_auto14} -\end{align} -$$ - -

    -This might look like a cumberstone to set up the correct expression -for finding the gradients. Luckily, Autograd comes to the rescue. - -

    - - -

    def solve_ode_neural_network(x, num_neurons_hidden, num_iter, lmb):
-    ## Set up initial weigths and biases 
-    
-    # For the hidden layer
-    p0 = npr.randn(num_neurons_hidden, 2 ) 
-
-    # For the output layer
-    p1 = npr.randn(1, num_neurons_hidden + 1 ) # +1 since bias is included
-
-    P = [p0, p1]
-
-    print('Initial cost: %g'%cost_function(P, x))
-    
-    ## Start finding the optimal weigths using gradient descent
-    
-    # Find the Python function that represents the gradient of the cost function
-    # w.r.t the 0-th input argument -- that is the weights and biases in the hidden and output layer
-    cost_function_grad = grad(cost_function,0)
-    
-    # Let the update be done num_iter times
-    for i in range(num_iter):
-        # Evaluate the gradient at the current weights and biases in P. 
-        # The cost_grad consist now of two arrays; 
-        # one for the gradient w.r.t P_hidden and 
-        # one for the gradient w.r.t P_output
-        cost_grad =  cost_function_grad(P, x)
-    
-        P[0] = P[0] - lmb * cost_grad[0]
-        P[1] = P[1] - lmb * cost_grad[1]
-
-    print('Final cost: %g'%cost_function(P, x))
-    
-    return P
-
    -

    -









    - -

    An implementation of a Deep Neural Network

    - -

    -As previously stated, a Deep Neural Network (DNN) follows the same -concept of a neural network, but having more than one hidden -layer. Suppose that the network has \( N_{\mathrm{hidden}} \) hidden layers -where the \( l \)-th layer has \( N_{\mathrm{hidden}}^{(l)} \) neurons. The -input is still assumed to be an array of size \( 1 \times N \). The -network must now try to optimalize its output with respect to the -collection of weigths and biases \( P = \big\{P_{\mathrm{input} }, \ -P_{\mathrm{hidden} }^{(1)}, \ P_{\mathrm{hidden} }^{(2)}, \ \dots , \ -P_{\mathrm{hidden} }^{(N_{\mathrm{hidden}})}, \ P_{\mathrm{output} }\big\} \). - -

    -









    - -

    The final parts of the code

    -

    - - -

    def deep_neural_network(deep_params, x):
-    # N_hidden is the number of hidden layers  
-    N_hidden = np.size(deep_params) - 1 # -1 since params consist of parameters to all the hidden layers AND the output layer
-        
-    # Assumes input x being an one-dimensional array
-    num_values = np.size(x)
-    x = x.reshape(-1, num_values)
-    
-    # Assume that the input layer does nothing to the input x
-    x_input = x
-    
-    # Due to multiple hidden layers, define a variable referencing to the
-    # output of the previous layer:
-    x_prev = x_input 
-    
-    ## Hidden layers:
-    
-    for l in range(N_hidden):
-        # From the list of parameters P; find the correct weigths and bias for this layer
-        w_hidden = deep_params[l]
-        
-        # Add a row of ones to include bias
-        x_prev = np.concatenate((np.ones((1,num_values)), x_prev ), axis = 0)
-
-        z_hidden = np.matmul(w_hidden, x_prev)
-        x_hidden = sigmoid(z_hidden)
-
-        # Update x_prev such that next layer can use the output from this layer
-        x_prev = x_hidden 
-
-    ## Output layer:
-    
-    # Get the weights and bias for this layer
-    w_output = deep_params[-1]
-    
-    # Include bias:
-    x_prev = np.concatenate((np.ones((1,num_values)), x_prev), axis = 0)
-
-    z_output = np.matmul(w_output, x_prev)
-    x_output = z_output
-
-    return x_output
-
    -

    -









    - -

    And adding Back propagation

    - -

    -This step is very similar for the neural network. The idea in this -step is the same as for the neural network, but with more parameters -to update for. Again there is no need for computing the gradients -analytically since Autograd does the work for us. - -

    - - -

    # The trial solution using the deep neural network:
-def g_trial_deep(x,params, g0 = 10):
-    return g0 + x*deep_neural_network(params,x)
-
-# The same cost function as for the neural network, but calls deep_neural_network instead.
-def cost_function_deep(P, x):
-    
-    # Evaluate the trial function with the current parameters P
-    g_t = g_trial_deep(x,P)
-    
-    # Find the derivative w.r.t x of the neural network
-    d_net_out = elementwise_grad(deep_neural_network,1)(P,x) 
-    
-    # Find the derivative w.r.t x of the trial function
-    d_g_t = elementwise_grad(g_trial_deep,0)(x,P)  
-    
-    # The right side of the ODE 
-    func = g(x, g_t)
-
-    err_sqr = (d_g_t - func)**2
-    cost_sum = np.sum(err_sqr)
-    
-    return cost_sum
-
-def solve_ode_deep_neural_network(x, num_neurons, num_iter, lmb):
-    # num_hidden_neurons is now a list of number of neurons within each hidden layer
-
-    # Find the number of hidden layers:
-    N_hidden = np.size(num_neurons)
-    
-    ## Set up initial weigths and biases 
-    
-    # Initialize the list of parameters:
-    P = [None]*(N_hidden + 1) # + 1 to include the output layer
-
-    P[0] = npr.randn(num_neurons[0], 2 ) 
-    for l in range(1,N_hidden):
-        P[l] = npr.randn(num_neurons[l], num_neurons[l-1] + 1) # +1 to include bias 
-    
-    # For the output layer
-    P[-1] = npr.randn(1, num_neurons[-1] + 1 ) # +1 since bias is included
-
-    print('Initial cost: %g'%cost_function_deep(P, x))
-    
-    ## Start finding the optimal weigths using gradient descent
-    
-    # Find the Python function that represents the gradient of the cost function
-    # w.r.t the 0-th input argument -- that is the weights and biases in the hidden and output layer
-    cost_function_deep_grad = grad(cost_function_deep,0)
-    
-    # Let the update be done num_iter times
-    for i in range(num_iter):
-        # Evaluate the gradient at the current weights and biases in P. 
-        # The cost_grad consist now of N_hidden + 1 arrays; the gradient w.r.t the weights and biases
-        # in the hidden layers and output layers evaluated at x.
-        cost_deep_grad =  cost_function_deep_grad(P, x)
-        
-        for l in range(N_hidden+1):
-            P[l] = P[l] - lmb * cost_deep_grad[l]
-
-    print('Final cost: %g'%cost_function_deep(P, x))
-    
-    return P
-
    -

    -









    - -

    Solving the ODE

    - -

    -Finally, having set up the networks we are ready to use them to solve the ODE problem. -We add the analytical solution - -

    - - -

    def g_analytic(x, gamma = 2, g0 = 10):
-    return g0*np.exp(-gamma*x)
-
    -

    -









    - -

    Using neural network

    - -

    -The code below solves the ODE using a neural network. The number of -values for the input \( \vec x \) is 10, number of hidden neurons in the -hidden layer being 10 and th step size used in gradien descent -\( \lambda = 0.001 \). The program updates the weights and biases in the -network for a given number of iterations. Finally, it plots the results from using the -neural network along with the analytical solution. - -

    - - -

    npr.seed(15)
-
-## Decide the vales of arguments to the function to solve
-N = 10
-x = np.linspace(0, 1, N)
-
-## Set up the initial parameters
-num_hidden_neurons = 10
-num_iter = 10000
-lmb = 0.001
-
-P = solve_ode_neural_network(x, num_hidden_neurons, num_iter, lmb)
-
-res = g_trial(x,P) 
-res_analytical = g_analytic(x)
-
-print('Max absolute difference: %g'%np.max(np.abs(res - res_analytical)))
-
-plt.figure(figsize=(10,10))
-
-plt.title('Performance of neural network solving an ODE compared to the analytical solution')
-plt.plot(x, res_analytical)
-plt.plot(x, res[0,:])
-plt.legend(['analytical','nn'])
-plt.xlabel('x')
-plt.ylabel('g(x)')
-plt.show()
-
    -

    - - -

    Using a deep neural network

    - -

    - - -

    npr.seed(15)
-
-## Decide the vales of arguments to the function to solve
-N = 10
-x = np.linspace(0, 1, N)
-
-## Set up the initial parameters
-num_hidden_neurons = np.array([10,10])
-num_iter = 10000
-lmb = 0.001
-
-P = solve_ode_deep_neural_network(x, num_hidden_neurons, num_iter, lmb)
-
-res = g_trial_deep(x,P) 
-res_analytical = g_analytic(x)
-
-plt.figure(figsize=(10,10))
-
-plt.title('Performance of a deep neural network solving an ODE compared to the analytical solution')
-plt.plot(x, res_analytical)
-plt.plot(x, res[0,:])
-plt.legend(['analytical','dnn'])
-plt.ylabel('g(x)')
-plt.show()
-
    -

    -









    - -

    Wrapping it up

    - -

    -By rewriting the ODE as a minimization problem, it was possible to -solve equation using either a neural network (one hidden layer) or a -deep neural network (more than one hidden layers). How well the -network performed is measured by a specified cost function, which is -the function the network tries to minimize. Using a trial solution -which satisfies the additional condition and being defined by using -the output from the network in some way, the minimization problem -could be explicitly defined for out network to solve. The proposed -solution from the network is then the trial solution with parameters, -that is weights and biases within each layer in the network, such that -the solution minimizes the cost function. diff --git a/doc/pub/NeuralNet/html/NeuralNet.html b/doc/pub/NeuralNet/html/NeuralNet.html index f7f414bdc..3d43606a9 100644 --- a/doc/pub/NeuralNet/html/NeuralNet.html +++ b/doc/pub/NeuralNet/html/NeuralNet.html @@ -7,9 +7,9 @@ Automatically generated HTML file from DocOnce source - + -Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning and convolutional networks +Data Analysis and Machine Learning: Neural networks, from the simple perceptron to deep learning + + + + + + + + + + + + + + + + + +

    +
    + + Convolutional Neural Networks +
    + +
    + +
    +
    +
    + +
    + +

     

     

     

    + + + + + + +
    +

    Convolutional Neural Networks

    + +

    + + +

    +Morten Hjorth-Jensen [1, 2] +
    + +

    + + +

    [1] Department of Physics, University of Oslo
    +
    [2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University
    +
    +

    +

    Oct 3, 2019

    +
    +

    + + +

    Read »

    + + +
    + +

    + +

    + + +
    + + + + + + + +
    + © 1999-2019, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license +
    + + + + + + diff --git a/doc/src/CNN/cnn-minted.pdf b/doc/src/CNN/cnn-minted.pdf new file mode 100644 index 0000000000000000000000000000000000000000..f2e9ec1f896a3e1a38cb25baf778bbfcb4ccfe73 GIT binary patch literal 336428 zcmd422UJv1vnG1T86@W*S&*EOAX!AB5+t<>NRo`ih9>7AqM#@_OK8cGGmRigq6Ep3 zp<7Z59j5*7y!rn-_uW}DcfB?1ZF;kI_c^C}*EzL!ef3qffQh!Qn3T92xxnPi-~zcK zk0g)RGdFS-6>^COPM*&GEA&ShA(pn+J|J-!uxgwni5+#9HWk z)<}*BC{%5Qh0Hclh)o+))O@e~kzVLt^}KE-))0#BQcu&HobR=js5!kPq7yX&>Q(#-PqTZX;CM-=I7vNc!dcm;-yk|^@R+G=**t$jy_L}7Tq?D z(Y+^{S;{0XN_GGWw>;lu5-%9>k4a|D4G26#UlK>sXjQ~#6aHzU%D?gJ`eurcPuJ#7 z;hwnkJA<2cdAD<43FfbE8Ly`U=Iwov_dzrct4%8bg&cJiWpcUpmLz<5@OHFkU5dh| z_JAfRY_isMd%w_AStYI+{b&Whh~S`ds1{CBU5M$k;ccp3TxukofBa&kiTZSB^M-y&y>A({ZhNc0H_P8g*-@_0=}t zM8-r&nlj=eDf-oq>dTicp8eeXQb3$X(*?opy1lVl9-Xo^x5xO^y<$nr0z-YwUw%j{ zd;&bzxV;-*?G3!D(~EOa;r)*06(L~N6qfdX8OE8YcXu|FgGKp!^bl_l>C)sQRk2h~ zy!LR$MK{f-&%9%`^ZZ_vc#qJHX3z#YYELp{a7ws+6fU`+BBJ5Zc<_9a`<_RJ7RQUK 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+%\usepackage{float}\restylefloat{figure} + +% --- fancyhdr package for fancy headers --- +\usepackage{fancyhdr} +\fancyhf{} % sets both header and footer to nothing +\renewcommand{\headrulewidth}{0pt} +\fancyfoot[LE,RO]{\thepage} +% Ensure copyright on titlepage (article style) and chapter pages (book style) +\fancypagestyle{plain}{ + \fancyhf{} + \fancyfoot[C]{{\footnotesize \copyright\ 1999-2019, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license}} +% \renewcommand{\footrulewidth}{0mm} + \renewcommand{\headrulewidth}{0mm} +} +% Ensure copyright on titlepages with \thispagestyle{empty} +\fancypagestyle{empty}{ + \fancyhf{} + \fancyfoot[C]{{\footnotesize \copyright\ 1999-2019, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license}} + \renewcommand{\footrulewidth}{0mm} + \renewcommand{\headrulewidth}{0mm} +} + +\pagestyle{fancy} + + +% prevent orhpans and widows +\clubpenalty = 10000 +\widowpenalty = 10000 + +% --- end of standard preamble for documents --- + + +% insert custom LaTeX commands... + +\raggedbottom +\makeindex +\usepackage[totoc]{idxlayout} % for index in the toc +\usepackage[nottoc]{tocbibind} % for references/bibliography in the toc + +%-------------------- end preamble ---------------------- + +\begin{document} + +% matching end for #ifdef PREAMBLE + +\newcommand{\exercisesection}[1]{\subsection*{#1}} + + +% ------------------- main content ---------------------- + + + +% ----------------- title ------------------------- + +\thispagestyle{empty} + +\begin{center} +{\LARGE\bf +\begin{spacing}{1.25} +Convolutional Neural Networks +\end{spacing} +} +\end{center} + +% ----------------- author(s) ------------------------- + +\begin{center} +{\bf Morten Hjorth-Jensen${}^{1, 2}$} \\ [0mm] +\end{center} + +\begin{center} +% List of all institutions: +\centerline{{\small ${}^1$Department of Physics, University of Oslo}} +\centerline{{\small ${}^2$Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University}} +\end{center} + +% ----------------- end author(s) ------------------------- + +% --- begin date --- +\begin{center} +Oct 3, 2019 +\end{center} +% --- end date --- + +\vspace{1cm} + + +% !split +\subsection*{Convolutional Neural Networks (recognizing images)} + +Convolutional Neural Networks (CNN) are very similar to ordinary Neural Networks. + +They are made up of neurons that have learnable weights and +biases. Each neuron receives some inputs, performs a dot product and +optionally follows it with a non-linearity. The whole network still +expresses a single differentiable score function: from the raw image +pixels on one end to class scores at the other. And they still have a +loss function (for example Softmax) on the last (fully-connected) layer +and all the tips/tricks we developed for learning regular Neural +Networks still apply (back propagation, gradient descent etc etc). + +What is the difference? \textbf{CNN architectures make the explicit assumption that +the inputs are images, which allows us to encode certain properties +into the architecture. These then make the forward function more +efficient to implement and vastly reduce the amount of parameters in +the network.} + +Here we provide only a superficial overview, for the more interested, we recommend highly the course +\href{{https://www.uio.no/studier/emner/matnat/ifi/IN5400/index-eng.html}}{IN5400 – Machine Learning for Image Analysis} +and the slides of \href{{http://cs231n.github.io/convolutional-networks/}}{CS231}. + +% !split +\subsection*{Regular NNs don’t scale well to full images} + +As an example, consider +an image of size $32\times 32\times 3$ (32 wide, 32 high, 3 color channels), so a +single fully-connected neuron in a first hidden layer of a regular +Neural Network would have $32\times 32\times 3 = 3072$ weights. This amount still +seems manageable, but clearly this fully-connected structure does not +scale to larger images. For example, an image of more respectable +size, say $200\times 200\times 3$, would lead to neurons that have +$200\times 200\times 3 = 120,000$ weights. + +We could have +several such neurons, and the parameters would add up quickly! Clearly, +this full connectivity is wasteful and the huge number of parameters +would quickly lead to possible overfitting. + + +\begin{figure}[!ht] % + \centerline{\includegraphics[width=0.6\linewidth]{figslides/nn.jpeg}} + \caption{ + A regular 3-layer Neural Network. + } +\end{figure} +%\clearpage % flush figures + + +% !split +\subsection*{3D volumes of neurons} + +Convolutional Neural Networks take advantage of the fact that the +input consists of images and they constrain the architecture in a more +sensible way. + +In particular, unlike a regular Neural Network, the +layers of a CNN have neurons arranged in 3 dimensions: width, +height, depth. (Note that the word depth here refers to the third +dimension of an activation volume, not to the depth of a full Neural +Network, which can refer to the total number of layers in a network.) + +To understand it better, the above example of an image +with an input volume of +activations has dimensions $32\times 32\times 3$ (width, height, +depth respectively). + +The neurons in a layer will +only be connected to a small region of the layer before it, instead of +all of the neurons in a fully-connected manner. Moreover, the final +output layer could for this specific image have dimensions $1\times 1 \times 10$, +because by the +end of the CNN architecture we will reduce the full image into a +single vector of class scores, arranged along the depth +dimension. + + +\begin{figure}[!ht] % + \centerline{\includegraphics[width=0.6\linewidth]{figslides/cnn.jpeg}} + \caption{ + A CNN arranges its neurons in three dimensions (width, height, depth), as visualized in one of the layers. Every layer of a CNN transforms the 3D input volume to a 3D output volume of neuron activations. In this example, the red input layer holds the image, so its width and height would be the dimensions of the image, and the depth would be 3 (Red, Green, Blue channels). + } +\end{figure} +%\clearpage % flush figures + + + + +% !split +\subsection*{Layers used to build CNNs} + + +A simple CNN is a sequence of layers, and every layer of a CNN +transforms one volume of activations to another through a +differentiable function. We use three main types of layers to build +CNN architectures: Convolutional Layer, Pooling Layer, and +Fully-Connected Layer (exactly as seen in regular Neural Networks). We +will stack these layers to form a full CNN architecture. + +A simple CNN for image classification could have the architecture: + +\begin{itemize} +\item \textbf{INPUT} ($32\times 32 \times 3$) will hold the raw pixel values of the image, in this case an image of width 32, height 32, and with three color channels R,G,B. + +\item \textbf{CONV} (convolutional )layer will compute the output of neurons that are connected to local regions in the input, each computing a dot product between their weights and a small region they are connected to in the input volume. This may result in volume such as $[32\times 32\times 12]$ if we decided to use 12 filters. + +\item \textbf{RELU} layer will apply an elementwise activation function, such as the $max(0,x)$ thresholding at zero. This leaves the size of the volume unchanged ($[32\times 32\times 12]$). + +\item \textbf{POOL} (pooling) layer will perform a downsampling operation along the spatial dimensions (width, height), resulting in volume such as $[16\times 16\times 12]$. + +\item \textbf{FC} (i.e.~fully-connected) layer will compute the class scores, resulting in volume of size $[1\times 1\times 10]$, where each of the 10 numbers correspond to a class score, such as among the 10 categories of the MNIST images we considered above . As with ordinary Neural Networks and as the name implies, each neuron in this layer will be connected to all the numbers in the previous volume. +\end{itemize} + +\noindent +% !split +\subsection*{Transforming images} + +CNNs transform the original image layer by layer from the original +pixel values to the final class scores. + +Observe that some layers contain +parameters and other don’t. In particular, the CNN layers perform +transformations that are a function of not only the activations in the +input volume, but also of the parameters (the weights and biases of +the neurons). On the other hand, the RELU/POOL layers will implement a +fixed function. The parameters in the CONV/FC layers will be trained +with gradient descent so that the class scores that the CNN computes +are consistent with the labels in the training set for each image. + + +% !split +\subsection*{CNNs in brief} + +In summary: + +\begin{itemize} +\item A CNN architecture is in the simplest case a list of Layers that transform the image volume into an output volume (e.g.~holding the class scores) + +\item There are a few distinct types of Layers (e.g.~CONV/FC/RELU/POOL are by far the most popular) + +\item Each Layer accepts an input 3D volume and transforms it to an output 3D volume through a differentiable function + +\item Each Layer may or may not have parameters (e.g.~CONV/FC do, RELU/POOL don’t) + +\item Each Layer may or may not have additional hyperparameters (e.g.~CONV/FC/POOL do, RELU doesn’t) +\end{itemize} + +\noindent +For more material on convolutional networks, we strongly recommend +the course +\href{{https://www.uio.no/studier/emner/matnat/ifi/IN5400/index-eng.html}}{IN5400 – Machine Learning for Image Analysis} +and the slides of \href{{http://cs231n.github.io/convolutional-networks/}}{CS231} which is taught at Stanford University (consistently ranked as one of the top computer science programs in the world). \href{{http://neuralnetworksanddeeplearning.com/chap6.html}}{Michael Nielsen's book is a must read, in particular chapter 6 which deals with CNNs}. + + +% !split +\subsection*{CNNs in more detail, building convolutional neural networks in Tensorflow and Keras} + + +As discussed above, CNNs are neural networks built from the assumption that the inputs +to the network are 2D images. This is important because the number of features or pixels in images +grows very fast with the image size, and an enormous number of weights and biases are needed in order to build an accurate network. + +As before, we still have our input, a hidden layer and an output. What's novel about convolutional networks +are the \textbf{convolutional} and \textbf{pooling} layers stacked in pairs between the input and the hidden layer. +In addition, the data is no longer represented as a 2D feature matrix, instead each input is a number of 2D +matrices, typically 1 for each color dimension (Red, Green, Blue). + + +% !split +\subsection*{Setting it up} + +It means that to represent the entire +dataset of images, we require a 4D matrix or \textbf{tensor}. This tensor has the dimensions: +\[ +(n_{inputs},\, n_{pixels, width},\, n_{pixels, height},\, depth) . +\] + +% !split +\subsection*{The MNIST dataset again} + +The MNIST dataset consists of grayscale images with a pixel size of +$28\times 28$, meaning we require $28 \times 28 = 724$ weights to each +neuron in the first hidden layer. + +If we were to analyze images of size $128\times 128$ we would require +$128 \times 128 = 16384$ weights to each neuron. Even worse if we were +dealing with color images, as most images are, we have an image matrix +of size $128\times 128$ for each color dimension (Red, Green, Blue), +meaning 3 times the number of weights $= 49152$ are required for every +single neuron in the first hidden layer. + + +% !split +\subsection*{Strong correlations} +Images typically have strong local correlations, meaning that a small +part of the image varies little from its neighboring regions. If for +example we have an image of a blue car, we can roughly assume that a +small blue part of the image is surrounded by other blue regions. + +Therefore, instead of connecting every single pixel to a neuron in the +first hidden layer, as we have previously done with deep neural +networks, we can instead connect each neuron to a small part of the +image (in all 3 RGB depth dimensions). The size of each small area is +fixed, and known as a \href{{https://en.wikipedia.org/wiki/Receptive_field}}{receptive}. + + +% !split +\subsection*{Layers of a CNN} +The layers of a convolutional neural network arrange neurons in 3D: width, height and depth. +The input image is typically a square matrix of depth 3. + +A \textbf{convolution} is performed on the image which outputs +a 3D volume of neurons. The weights to the input are arranged in a number of 2D matrices, known as \textbf{filters}. + + +Each filter slides along the input image, taking the dot product +between each small part of the image and the filter, in all depth +dimensions. This is then passed through a non-linear function, +typically the \textbf{Rectified Linear (ReLu)} function, which serves as the +activation of the neurons in the first convolutional layer. This is +further passed through a \textbf{pooling layer}, which reduces the size of the +convolutional layer, e.g.~by taking the maximum or average across some +small regions, and this serves as input to the next convolutional +layer. + + +% !split +\subsection*{Systematic reduction} + +By systematically reducing the size of the input volume, through +convolution and pooling, the network should create representations of +small parts of the input, and then from them assemble representations +of larger areas. The final pooling layer is flattened to serve as +input to a hidden layer, such that each neuron in the final pooling +layer is connected to every single neuron in the hidden layer. This +then serves as input to the output layer, e.g.~a softmax output for +classification. + + +% !split +\subsection*{Prerequisites: Collect and pre-process data} +\begin{minted}[fontsize=\fontsize{9pt}{9pt},linenos=false,mathescape,baselinestretch=1.0,fontfamily=tt,xleftmargin=7mm]{python} +# import necessary packages +import numpy as np +import matplotlib.pyplot as plt +from sklearn import datasets + + +# ensure the same random numbers appear every time +np.random.seed(0) + +# display images in notebook +%matplotlib inline +plt.rcParams['figure.figsize'] = (12,12) + + +# download MNIST dataset +digits = datasets.load_digits() + +# define inputs and labels +inputs = digits.images +labels = digits.target + +# RGB images have a depth of 3 +# our images are grayscale so they should have a depth of 1 +inputs = inputs[:,:,:,np.newaxis] + +print("inputs = (n_inputs, pixel_width, pixel_height, depth) = " + str(inputs.shape)) +print("labels = (n_inputs) = " + str(labels.shape)) + + +# choose some random images to display +n_inputs = len(inputs) +indices = np.arange(n_inputs) +random_indices = np.random.choice(indices, size=5) + +for i, image in enumerate(digits.images[random_indices]): + plt.subplot(1, 5, i+1) + plt.axis('off') + plt.imshow(image, cmap=plt.cm.gray_r, interpolation='nearest') + plt.title("Label: %d" % digits.target[random_indices[i]]) +plt.show() +\end{minted} + + +% !split +\subsection*{Importing Keras and Tensorflow} +\begin{minted}[fontsize=\fontsize{9pt}{9pt},linenos=false,mathescape,baselinestretch=1.0,fontfamily=tt,xleftmargin=7mm]{python} +from keras.utils import to_categorical +from sklearn.model_selection import train_test_split + +# representation of labels +labels = to_categorical(labels) + +# split into train and test data +# one-liner from scikit-learn library +train_size = 0.8 +test_size = 1 - train_size +X_train, X_test, Y_train, Y_test = train_test_split(inputs, labels, train_size=train_size, + test_size=test_size) +\end{minted} + +% !split +\subsection*{Using TensorFlow backend} + +We need to define model and architecture and choose cost function and optmizer. +\begin{Verbatim}[numbers=none,fontsize=\fontsize{9pt}{9pt},baselinestretch=0.95] + +import tensorflow as tf + +class ConvolutionalNeuralNetworkTensorflow: + def __init__( + self, + X_train, + Y_train, + X_test, + Y_test, + n_filters=10, + n_neurons_connected=50, + n_categories=10, + receptive_field=3, + stride=1, + padding=1, + epochs=10, + batch_size=100, + eta=0.1, + lmbd=0.0): + + self.global_step = tf.Variable(0, dtype=tf.int32, trainable=False, name='global_step') + + self.X_train = X_train + self.Y_train = Y_train + self.X_test = X_test + self.Y_test = Y_test + + self.n_inputs, self.input_width, self.input_height, self.depth = X_train.shape + + self.n_filters = n_filters + self.n_downsampled = int(self.input_width*self.input_height*n_filters / 4) + self.n_neurons_connected = n_neurons_connected + self.n_categories = n_categories + + self.receptive_field = receptive_field + self.stride = stride + self.strides = [stride, stride, stride, stride] + self.padding = padding + + self.epochs = epochs + self.batch_size = batch_size + self.iterations = self.n_inputs // self.batch_size + self.eta = eta + self.lmbd = lmbd + + self.create_placeholders() + self.create_CNN() + self.create_loss() + self.create_optimiser() + self.create_accuracy() + + def create_placeholders(self): + with tf.name_scope('data'): + self.X = tf.placeholder(tf.float32, shape=(None, self.input_width, self.input_height, self.depth), name='X_data') + self.Y = tf.placeholder(tf.float32, shape=(None, self.n_categories), name='Y_data') + + def create_CNN(self): + with tf.name_scope('CNN'): + + # Convolutional layer + self.W_conv = self.weight_variable([self.receptive_field, self.receptive_field, self.depth, self.n_filters], name='conv', dtype=tf.float32) + b_conv = self.weight_variable([self.n_filters], name='conv', dtype=tf.float32) + z_conv = tf.nn.conv2d(self.X, self.W_conv, self.strides, padding='SAME', name='conv') + b_conv + a_conv = tf.nn.relu(z_conv) + + # 2x2 max pooling + a_pool = tf.nn.max_pool(a_conv, [1, 2, 2, 1], [1, 2, 2, 1], padding='SAME', name='pool') + + # Fully connected layer + a_pool_flat = tf.reshape(a_pool, [-1, self.n_downsampled]) + self.W_fc = self.weight_variable([self.n_downsampled, self.n_neurons_connected], name='fc', dtype=tf.float32) + b_fc = self.bias_variable([self.n_neurons_connected], name='fc', dtype=tf.float32) + a_fc = tf.nn.relu(tf.matmul(a_pool_flat, self.W_fc) + b_fc) + + # Output layer + self.W_out = self.weight_variable([self.n_neurons_connected, self.n_categories], name='out', dtype=tf.float32) + b_out = self.bias_variable([self.n_categories], name='out', dtype=tf.float32) + self.z_out = tf.matmul(a_fc, self.W_out) + b_out + + def create_loss(self): + with tf.name_scope('loss'): + softmax_loss = tf.reduce_mean(tf.nn.softmax_cross_entropy_with_logits_v2(labels=self.Y, logits=self.z_out)) + + regularizer_loss_conv = tf.nn.l2_loss(self.W_conv) + regularizer_loss_fc = tf.nn.l2_loss(self.W_fc) + regularizer_loss_out = tf.nn.l2_loss(self.W_out) + regularizer_loss = self.lmbd*(regularizer_loss_conv + regularizer_loss_fc + regularizer_loss_out) + + self.loss = softmax_loss + regularizer_loss + + def create_accuracy(self): + with tf.name_scope('accuracy'): + probabilities = tf.nn.softmax(self.z_out) + predictions = tf.argmax(probabilities, 1) + labels = tf.argmax(self.Y, 1) + + correct_predictions = tf.equal(predictions, labels) + correct_predictions = tf.cast(correct_predictions, tf.float32) + self.accuracy = tf.reduce_mean(correct_predictions) + + def create_optimiser(self): + with tf.name_scope('optimizer'): + self.optimizer = tf.train.GradientDescentOptimizer(learning_rate=self.eta).minimize(self.loss, global_step=self.global_step) + + def weight_variable(self, shape, name='', dtype=tf.float32): + initial = tf.truncated_normal(shape, stddev=0.1) + return tf.Variable(initial, name=name, dtype=dtype) + + def bias_variable(self, shape, name='', dtype=tf.float32): + initial = tf.constant(0.1, shape=shape) + return tf.Variable(initial, name=name, dtype=dtype) + + def fit(self): + data_indices = np.arange(self.n_inputs) + + with tf.Session() as sess: + sess.run(tf.global_variables_initializer()) + for i in range(self.epochs): + for j in range(self.iterations): + chosen_datapoints = np.random.choice(data_indices, size=self.batch_size, replace=False) + batch_X, batch_Y = self.X_train[chosen_datapoints], self.Y_train[chosen_datapoints] + + sess.run([CNN.loss, CNN.optimizer], + feed_dict={CNN.X: batch_X, + CNN.Y: batch_Y}) + accuracy = sess.run(CNN.accuracy, + feed_dict={CNN.X: batch_X, + CNN.Y: batch_Y}) + step = sess.run(CNN.global_step) + + self.train_loss, self.train_accuracy = sess.run([CNN.loss, CNN.accuracy], + feed_dict={CNN.X: self.X_train, + CNN.Y: self.Y_train}) + + self.test_loss, self.test_accuracy = sess.run([CNN.loss, CNN.accuracy], + feed_dict={CNN.X: self.X_test, + CNN.Y: self.Y_test}) +\end{Verbatim} + +% !split +\subsection*{Train the model} + +We need now to train the model, evaluate it and test its performance on test data, and eventually include hyperparameters. +\begin{minted}[fontsize=\fontsize{9pt}{9pt},linenos=false,mathescape,baselinestretch=1.0,fontfamily=tt,xleftmargin=7mm]{python} +epochs = 100 +batch_size = 100 +n_filters = 10 +n_neurons_connected = 50 +n_categories = 10 + +eta_vals = np.logspace(-5, 1, 7) +lmbd_vals = np.logspace(-5, 1, 7) +CNN_tf = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object) + +for i, eta in enumerate(eta_vals): + for j, lmbd in enumerate(lmbd_vals): + CNN = ConvolutionalNeuralNetworkTensorflow(X_train, Y_train, X_test, Y_test, + n_filters=n_filters, n_neurons_connected=n_neurons_connected, + n_categories=n_categories, epochs=epochs, batch_size=batch_size, + eta=eta, lmbd=lmbd) + CNN.fit() + + print("Learning rate = ", eta) + print("Lambda = ", lmbd) + print("Test accuracy: %.3f" % CNN.test_accuracy) + print() + + CNN_tf[i][j] = CNN +\end{minted} + +% !split +\subsection*{Visualizing the results} + +\begin{minted}[fontsize=\fontsize{9pt}{9pt},linenos=false,mathescape,baselinestretch=1.0,fontfamily=tt,xleftmargin=7mm]{python} +# visual representation of grid search +# uses seaborn heatmap, could probably do this in matplotlib +import seaborn as sns + +sns.set() + +train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals))) +test_accuracy = np.zeros((len(eta_vals), len(lmbd_vals))) + +for i in range(len(eta_vals)): + for j in range(len(lmbd_vals)): + CNN = CNN_tf[i][j] + + train_accuracy[i][j] = CNN.train_accuracy + test_accuracy[i][j] = CNN.test_accuracy + + +fig, ax = plt.subplots(figsize = (10, 10)) +sns.heatmap(train_accuracy, annot=True, ax=ax, cmap="viridis") +ax.set_title("Training Accuracy") +ax.set_ylabel("$\eta$") +ax.set_xlabel("$\lambda$") +plt.show() + +fig, ax = plt.subplots(figsize = (10, 10)) +sns.heatmap(test_accuracy, annot=True, ax=ax, cmap="viridis") +ax.set_title("Test Accuracy") +ax.set_ylabel("$\eta$") +ax.set_xlabel("$\lambda$") +plt.show() +\end{minted} + +% !split +\subsection*{Running with Keras} + +\begin{minted}[fontsize=\fontsize{9pt}{9pt},linenos=false,mathescape,baselinestretch=1.0,fontfamily=tt,xleftmargin=7mm]{python} +from keras.models import Sequential +from keras.layers.convolutional import Conv2D +from keras.layers.convolutional import MaxPooling2D +from keras.layers import Flatten +from keras.layers import Dense +from keras.regularizers import l2 +from keras.optimizers import SGD + +def create_convolutional_neural_network_keras(input_shape, receptive_field, + n_filters, n_neurons_connected, n_categories, + eta, lmbd): + model = Sequential() + model.add(Conv2D(n_filters, (receptive_field, receptive_field), input_shape=input_shape, padding='same', + activation='relu', kernel_regularizer=l2(lmbd))) + model.add(MaxPooling2D(pool_size=(2, 2))) + model.add(Flatten()) + model.add(Dense(n_neurons_connected, activation='relu', kernel_regularizer=l2(lmbd))) + model.add(Dense(n_categories, activation='softmax', kernel_regularizer=l2(lmbd))) + + sgd = SGD(lr=eta) + model.compile(loss='categorical_crossentropy', optimizer=sgd, metrics=['accuracy']) + + return model + +epochs = 100 +batch_size = 100 +input_shape = X_train.shape[1:4] +receptive_field = 3 +n_filters = 10 +n_neurons_connected = 50 +n_categories = 10 + +eta_vals = np.logspace(-5, 1, 7) +lmbd_vals = np.logspace(-5, 1, 7) +\end{minted} + +% !split +\subsection*{Final part} + +\begin{minted}[fontsize=\fontsize{9pt}{9pt},linenos=false,mathescape,baselinestretch=1.0,fontfamily=tt,xleftmargin=7mm]{python} +CNN_keras = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object) + +for i, eta in enumerate(eta_vals): + for j, lmbd in enumerate(lmbd_vals): + CNN = create_convolutional_neural_network_keras(input_shape, receptive_field, + n_filters, n_neurons_connected, n_categories, + eta, lmbd) + CNN.fit(X_train, Y_train, epochs=epochs, batch_size=batch_size, verbose=0) + scores = CNN.evaluate(X_test, Y_test) + + CNN_keras[i][j] = CNN + + print("Learning rate = ", eta) + print("Lambda = ", lmbd) + print("Test accuracy: %.3f" % scores[1]) + print() +\end{minted} + +% !split +\subsection*{Final visualization} + +\begin{Verbatim}[numbers=none,fontsize=\fontsize{9pt}{9pt},baselinestretch=0.95] +# visual representation of grid search +# uses seaborn heatmap, could probably do this in matplotlib +import seaborn as sns + +sns.set() + +train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals))) +test_accuracy = np.zeros((len(eta_vals), len(lmbd_vals))) + +for i in range(len(eta_vals)): + for j in range(len(lmbd_vals)): + CNN = CNN_keras[i][j] + + train_accuracy[i][j] = CNN.evaluate(X_train, Y_train)[1] + test_accuracy[i][j] = CNN.evaluate(X_test, Y_test)[1] + + +fig, ax = plt.subplots(figsize = (10, 10)) +sns.heatmap(train_accuracy, annot=True, ax=ax, cmap="viridis") +ax.set_title("Training Accuracy") +ax.set_ylabel("$\eta$") +ax.set_xlabel("$\lambda$") +plt.show() + +fig, ax = plt.subplots(figsize = (10, 10)) +sns.heatmap(test_accuracy, annot=True, ax=ax, cmap="viridis") +ax.set_title("Test Accuracy") +ax.set_ylabel("$\eta$") +ax.set_xlabel("$\lambda$") +plt.show() +\end{Verbatim} + +% !split +\subsection*{Fun links} + +\begin{enumerate} +\item \href{{https://arxiv.org/abs/1604.07316}}{Self-Driving cars using a convolutional neural network} + +\item \href{{https://deepdreamgenerator.com/}}{Abstract art using convolutional neural networks} +\end{enumerate} + +\noindent + +% ------------------- end of main content --------------- + +\end{document} + diff --git a/doc/src/CNN/cnn-reveal.html b/doc/src/CNN/cnn-reveal.html new file mode 100644 index 000000000..78ea52651 --- /dev/null +++ b/doc/src/CNN/cnn-reveal.html @@ -0,0 +1,984 @@ + + + + + + + +Convolutional Neural Networks + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

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    Convolutional Neural Networks

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    +Morten Hjorth-Jensen [1, 2] +
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    [1] Department of Physics, University of Oslo
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    [2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University
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    Oct 3, 2019

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    + © 1999-2019, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license +
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    Convolutional Neural Networks (recognizing images)

    + +

    +Convolutional Neural Networks (CNN) are very similar to ordinary Neural Networks. + +

    +They are made up of neurons that have learnable weights and +biases. Each neuron receives some inputs, performs a dot product and +optionally follows it with a non-linearity. The whole network still +expresses a single differentiable score function: from the raw image +pixels on one end to class scores at the other. And they still have a +loss function (for example Softmax) on the last (fully-connected) layer +and all the tips/tricks we developed for learning regular Neural +Networks still apply (back propagation, gradient descent etc etc). + +

    +What is the difference? CNN architectures make the explicit assumption that +the inputs are images, which allows us to encode certain properties +into the architecture. These then make the forward function more +efficient to implement and vastly reduce the amount of parameters in +the network. + +

    +Here we provide only a superficial overview, for the more interested, we recommend highly the course +IN5400 – Machine Learning for Image Analysis +and the slides of CS231. +

    + + +
    +

    Regular NNs don’t scale well to full images

    + +

    +As an example, consider +an image of size \( 32\times 32\times 3 \) (32 wide, 32 high, 3 color channels), so a +single fully-connected neuron in a first hidden layer of a regular +Neural Network would have \( 32\times 32\times 3 = 3072 \) weights. This amount still +seems manageable, but clearly this fully-connected structure does not +scale to larger images. For example, an image of more respectable +size, say \( 200\times 200\times 3 \), would lead to neurons that have +\( 200\times 200\times 3 = 120,000 \) weights. + +

    +We could have +several such neurons, and the parameters would add up quickly! Clearly, +this full connectivity is wasteful and the huge number of parameters +would quickly lead to possible overfitting. + +

    +

    +
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    Figure 1: A regular 3-layer Neural Network.

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    3D volumes of neurons

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    +Convolutional Neural Networks take advantage of the fact that the +input consists of images and they constrain the architecture in a more +sensible way. + +

    +In particular, unlike a regular Neural Network, the +layers of a CNN have neurons arranged in 3 dimensions: width, +height, depth. (Note that the word depth here refers to the third +dimension of an activation volume, not to the depth of a full Neural +Network, which can refer to the total number of layers in a network.) + +

    +To understand it better, the above example of an image +with an input volume of +activations has dimensions \( 32\times 32\times 3 \) (width, height, +depth respectively). + +

    +The neurons in a layer will +only be connected to a small region of the layer before it, instead of +all of the neurons in a fully-connected manner. Moreover, the final +output layer could for this specific image have dimensions \( 1\times 1 \times 10 \), +because by the +end of the CNN architecture we will reduce the full image into a +single vector of class scores, arranged along the depth +dimension. + +

    +

    +
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    Figure 2: A CNN arranges its neurons in three dimensions (width, height, depth), as visualized in one of the layers. Every layer of a CNN transforms the 3D input volume to a 3D output volume of neuron activations. In this example, the red input layer holds the image, so its width and height would be the dimensions of the image, and the depth would be 3 (Red, Green, Blue channels).

    +

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    Layers used to build CNNs

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    +A simple CNN is a sequence of layers, and every layer of a CNN +transforms one volume of activations to another through a +differentiable function. We use three main types of layers to build +CNN architectures: Convolutional Layer, Pooling Layer, and +Fully-Connected Layer (exactly as seen in regular Neural Networks). We +will stack these layers to form a full CNN architecture. + +

    +A simple CNN for image classification could have the architecture: + +

      +

    • INPUT (\( 32\times 32 \times 3 \)) will hold the raw pixel values of the image, in this case an image of width 32, height 32, and with three color channels R,G,B.
      • +

    • CONV (convolutional )layer will compute the output of neurons that are connected to local regions in the input, each computing a dot product between their weights and a small region they are connected to in the input volume. This may result in volume such as \( [32\times 32\times 12] \) if we decided to use 12 filters.
      • +

    • RELU layer will apply an elementwise activation function, such as the \( max(0,x) \) thresholding at zero. This leaves the size of the volume unchanged (\( [32\times 32\times 12] \)).
      • +

    • POOL (pooling) layer will perform a downsampling operation along the spatial dimensions (width, height), resulting in volume such as \( [16\times 16\times 12] \).
      • +

    • FC (i.e. fully-connected) layer will compute the class scores, resulting in volume of size \( [1\times 1\times 10] \), where each of the 10 numbers correspond to a class score, such as among the 10 categories of the MNIST images we considered above . As with ordinary Neural Networks and as the name implies, each neuron in this layer will be connected to all the numbers in the previous volume.
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    Transforming images

    + +

    +CNNs transform the original image layer by layer from the original +pixel values to the final class scores. + +

    +Observe that some layers contain +parameters and other don’t. In particular, the CNN layers perform +transformations that are a function of not only the activations in the +input volume, but also of the parameters (the weights and biases of +the neurons). On the other hand, the RELU/POOL layers will implement a +fixed function. The parameters in the CONV/FC layers will be trained +with gradient descent so that the class scores that the CNN computes +are consistent with the labels in the training set for each image. +

    + + +
    +

    CNNs in brief

    + +

    +In summary: + +

      +

    • A CNN architecture is in the simplest case a list of Layers that transform the image volume into an output volume (e.g. holding the class scores)
      • +

    • There are a few distinct types of Layers (e.g. CONV/FC/RELU/POOL are by far the most popular)
      • +

    • Each Layer accepts an input 3D volume and transforms it to an output 3D volume through a differentiable function
      • +

    • Each Layer may or may not have parameters (e.g. CONV/FC do, RELU/POOL don’t)
      • +

    • Each Layer may or may not have additional hyperparameters (e.g. CONV/FC/POOL do, RELU doesn’t)
      • +
    +

    + +For more material on convolutional networks, we strongly recommend +the course +IN5400 – Machine Learning for Image Analysis +and the slides of CS231 which is taught at Stanford University (consistently ranked as one of the top computer science programs in the world). Michael Nielsen's book is a must read, in particular chapter 6 which deals with CNNs. +

    + + +
    +

    CNNs in more detail, building convolutional neural networks in Tensorflow and Keras

    + +

    +As discussed above, CNNs are neural networks built from the assumption that the inputs +to the network are 2D images. This is important because the number of features or pixels in images +grows very fast with the image size, and an enormous number of weights and biases are needed in order to build an accurate network. + +

    +As before, we still have our input, a hidden layer and an output. What's novel about convolutional networks +are the convolutional and pooling layers stacked in pairs between the input and the hidden layer. +In addition, the data is no longer represented as a 2D feature matrix, instead each input is a number of 2D +matrices, typically 1 for each color dimension (Red, Green, Blue). +

    + + +
    +

    Setting it up

    + +

    +It means that to represent the entire +dataset of images, we require a 4D matrix or tensor. This tensor has the dimensions: + +

     
    +$$ +(n_{inputs},\, n_{pixels, width},\, n_{pixels, height},\, depth) . +$$ +

     
    +

    + + +
    +

    The MNIST dataset again

    + +

    +The MNIST dataset consists of grayscale images with a pixel size of +\( 28\times 28 \), meaning we require \( 28 \times 28 = 724 \) weights to each +neuron in the first hidden layer. + +

    +If we were to analyze images of size \( 128\times 128 \) we would require +\( 128 \times 128 = 16384 \) weights to each neuron. Even worse if we were +dealing with color images, as most images are, we have an image matrix +of size \( 128\times 128 \) for each color dimension (Red, Green, Blue), +meaning 3 times the number of weights \( = 49152 \) are required for every +single neuron in the first hidden layer. +

    + + +
    +

    Strong correlations

    +Images typically have strong local correlations, meaning that a small +part of the image varies little from its neighboring regions. If for +example we have an image of a blue car, we can roughly assume that a +small blue part of the image is surrounded by other blue regions. + +

    +Therefore, instead of connecting every single pixel to a neuron in the +first hidden layer, as we have previously done with deep neural +networks, we can instead connect each neuron to a small part of the +image (in all 3 RGB depth dimensions). The size of each small area is +fixed, and known as a receptive. +

    + + +
    +

    Layers of a CNN

    +The layers of a convolutional neural network arrange neurons in 3D: width, height and depth. +The input image is typically a square matrix of depth 3. + +

    +A convolution is performed on the image which outputs +a 3D volume of neurons. The weights to the input are arranged in a number of 2D matrices, known as filters. + +

    +Each filter slides along the input image, taking the dot product +between each small part of the image and the filter, in all depth +dimensions. This is then passed through a non-linear function, +typically the Rectified Linear (ReLu) function, which serves as the +activation of the neurons in the first convolutional layer. This is +further passed through a pooling layer, which reduces the size of the +convolutional layer, e.g. by taking the maximum or average across some +small regions, and this serves as input to the next convolutional +layer. +

    + + +
    +

    Systematic reduction

    + +

    +By systematically reducing the size of the input volume, through +convolution and pooling, the network should create representations of +small parts of the input, and then from them assemble representations +of larger areas. The final pooling layer is flattened to serve as +input to a hidden layer, such that each neuron in the final pooling +layer is connected to every single neuron in the hidden layer. This +then serves as input to the output layer, e.g. a softmax output for +classification. +

    + + +
    +

    Prerequisites: Collect and pre-process data

    +

    + + +

    # import necessary packages
+import numpy as np
+import matplotlib.pyplot as plt
+from sklearn import datasets
+
+
+# ensure the same random numbers appear every time
+np.random.seed(0)
+
+# display images in notebook
+%matplotlib inline
+plt.rcParams['figure.figsize'] = (12,12)
+
+
+# download MNIST dataset
+digits = datasets.load_digits()
+
+# define inputs and labels
+inputs = digits.images
+labels = digits.target
+
+# RGB images have a depth of 3
+# our images are grayscale so they should have a depth of 1
+inputs = inputs[:,:,:,np.newaxis]
+
+print("inputs = (n_inputs, pixel_width, pixel_height, depth) = " + str(inputs.shape))
+print("labels = (n_inputs) = " + str(labels.shape))
+
+
+# choose some random images to display
+n_inputs = len(inputs)
+indices = np.arange(n_inputs)
+random_indices = np.random.choice(indices, size=5)
+
+for i, image in enumerate(digits.images[random_indices]):
+    plt.subplot(1, 5, i+1)
+    plt.axis('off')
+    plt.imshow(image, cmap=plt.cm.gray_r, interpolation='nearest')
+    plt.title("Label: %d" % digits.target[random_indices[i]])
+plt.show()
+
    +
    + + +
    +

    Importing Keras and Tensorflow

    +

    + + +

    from keras.utils import to_categorical
+from sklearn.model_selection import train_test_split
+
+# representation of labels
+labels = to_categorical(labels)
+
+# split into train and test data
+# one-liner from scikit-learn library
+train_size = 0.8
+test_size = 1 - train_size
+X_train, X_test, Y_train, Y_test = train_test_split(inputs, labels, train_size=train_size,
+                                                    test_size=test_size)
+
    +
    + + +
    +

    Using TensorFlow backend

    + +

    +We need to define model and architecture and choose cost function and optmizer. +

    + + +

    import tensorflow as tf
+
+class ConvolutionalNeuralNetworkTensorflow:
+    def __init__(
+            self,
+            X_train,
+            Y_train,
+            X_test,
+            Y_test,
+            n_filters=10,
+            n_neurons_connected=50,
+            n_categories=10,
+            receptive_field=3,
+            stride=1,
+            padding=1,
+            epochs=10,
+            batch_size=100,
+            eta=0.1,
+            lmbd=0.0):
+        
+        self.global_step = tf.Variable(0, dtype=tf.int32, trainable=False, name='global_step')
+        
+        self.X_train = X_train
+        self.Y_train = Y_train
+        self.X_test = X_test
+        self.Y_test = Y_test
+        
+        self.n_inputs, self.input_width, self.input_height, self.depth = X_train.shape
+        
+        self.n_filters = n_filters
+        self.n_downsampled = int(self.input_width*self.input_height*n_filters / 4)
+        self.n_neurons_connected = n_neurons_connected
+        self.n_categories = n_categories
+        
+        self.receptive_field = receptive_field
+        self.stride = stride
+        self.strides = [stride, stride, stride, stride]
+        self.padding = padding
+        
+        self.epochs = epochs
+        self.batch_size = batch_size
+        self.iterations = self.n_inputs // self.batch_size
+        self.eta = eta
+        self.lmbd = lmbd
+        
+        self.create_placeholders()
+        self.create_CNN()
+        self.create_loss()
+        self.create_optimiser()
+        self.create_accuracy()
+    
+    def create_placeholders(self):
+        with tf.name_scope('data'):
+            self.X = tf.placeholder(tf.float32, shape=(None, self.input_width, self.input_height, self.depth), name='X_data')
+            self.Y = tf.placeholder(tf.float32, shape=(None, self.n_categories), name='Y_data')
+    
+    def create_CNN(self):
+        with tf.name_scope('CNN'):
+            
+            # Convolutional layer
+            self.W_conv = self.weight_variable([self.receptive_field, self.receptive_field, self.depth, self.n_filters], name='conv', dtype=tf.float32)
+            b_conv = self.weight_variable([self.n_filters], name='conv', dtype=tf.float32)
+            z_conv = tf.nn.conv2d(self.X, self.W_conv, self.strides, padding='SAME', name='conv') + b_conv
+            a_conv = tf.nn.relu(z_conv)
+            
+            # 2x2 max pooling
+            a_pool = tf.nn.max_pool(a_conv, [1, 2, 2, 1], [1, 2, 2, 1], padding='SAME', name='pool')
+            
+            # Fully connected layer
+            a_pool_flat = tf.reshape(a_pool, [-1, self.n_downsampled])
+            self.W_fc = self.weight_variable([self.n_downsampled, self.n_neurons_connected], name='fc', dtype=tf.float32)
+            b_fc = self.bias_variable([self.n_neurons_connected], name='fc', dtype=tf.float32)
+            a_fc = tf.nn.relu(tf.matmul(a_pool_flat, self.W_fc) + b_fc)
+            
+            # Output layer
+            self.W_out = self.weight_variable([self.n_neurons_connected, self.n_categories], name='out', dtype=tf.float32)
+            b_out = self.bias_variable([self.n_categories], name='out', dtype=tf.float32)
+            self.z_out = tf.matmul(a_fc, self.W_out) + b_out
+    
+    def create_loss(self):
+        with tf.name_scope('loss'):
+            softmax_loss = tf.reduce_mean(tf.nn.softmax_cross_entropy_with_logits_v2(labels=self.Y, logits=self.z_out))
+            
+            regularizer_loss_conv = tf.nn.l2_loss(self.W_conv)
+            regularizer_loss_fc = tf.nn.l2_loss(self.W_fc)
+            regularizer_loss_out = tf.nn.l2_loss(self.W_out)
+            regularizer_loss = self.lmbd*(regularizer_loss_conv + regularizer_loss_fc + regularizer_loss_out)
+            
+            self.loss = softmax_loss + regularizer_loss
+
+    def create_accuracy(self):
+        with tf.name_scope('accuracy'):
+            probabilities = tf.nn.softmax(self.z_out)
+            predictions = tf.argmax(probabilities, 1)
+            labels = tf.argmax(self.Y, 1)
+            
+            correct_predictions = tf.equal(predictions, labels)
+            correct_predictions = tf.cast(correct_predictions, tf.float32)
+            self.accuracy = tf.reduce_mean(correct_predictions)
+    
+    def create_optimiser(self):
+        with tf.name_scope('optimizer'):
+            self.optimizer = tf.train.GradientDescentOptimizer(learning_rate=self.eta).minimize(self.loss, global_step=self.global_step)
+            
+    def weight_variable(self, shape, name='', dtype=tf.float32):
+        initial = tf.truncated_normal(shape, stddev=0.1)
+        return tf.Variable(initial, name=name, dtype=dtype)
+    
+    def bias_variable(self, shape, name='', dtype=tf.float32):
+        initial = tf.constant(0.1, shape=shape)
+        return tf.Variable(initial, name=name, dtype=dtype)
+
+    def fit(self):
+        data_indices = np.arange(self.n_inputs)
+
+        with tf.Session() as sess:
+            sess.run(tf.global_variables_initializer())
+            for i in range(self.epochs):
+                for j in range(self.iterations):
+                    chosen_datapoints = np.random.choice(data_indices, size=self.batch_size, replace=False)
+                    batch_X, batch_Y = self.X_train[chosen_datapoints], self.Y_train[chosen_datapoints]
+            
+                    sess.run([CNN.loss, CNN.optimizer],
+                        feed_dict={CNN.X: batch_X,
+                                   CNN.Y: batch_Y})
+                    accuracy = sess.run(CNN.accuracy,
+                        feed_dict={CNN.X: batch_X,
+                                   CNN.Y: batch_Y})
+                    step = sess.run(CNN.global_step)
+    
+            self.train_loss, self.train_accuracy = sess.run([CNN.loss, CNN.accuracy],
+                feed_dict={CNN.X: self.X_train,
+                           CNN.Y: self.Y_train})
+        
+            self.test_loss, self.test_accuracy = sess.run([CNN.loss, CNN.accuracy],
+                feed_dict={CNN.X: self.X_test,
+                           CNN.Y: self.Y_test})
+
    +
    + + +
    +

    Train the model

    + +

    +We need now to train the model, evaluate it and test its performance on test data, and eventually include hyperparameters. +

    + + +

    epochs = 100
+batch_size = 100
+n_filters = 10
+n_neurons_connected = 50
+n_categories = 10
+
+eta_vals = np.logspace(-5, 1, 7)
+lmbd_vals = np.logspace(-5, 1, 7)
+CNN_tf = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)
+        
+for i, eta in enumerate(eta_vals):
+    for j, lmbd in enumerate(lmbd_vals):
+        CNN = ConvolutionalNeuralNetworkTensorflow(X_train, Y_train, X_test, Y_test,
+                                      n_filters=n_filters, n_neurons_connected=n_neurons_connected,
+                                      n_categories=n_categories, epochs=epochs, batch_size=batch_size,
+                                      eta=eta, lmbd=lmbd)
+        CNN.fit()
+        
+        print("Learning rate = ", eta)
+        print("Lambda = ", lmbd)
+        print("Test accuracy: %.3f" % CNN.test_accuracy)
+        print()
+            
+        CNN_tf[i][j] = CNN
+
    +
    + + +
    +

    Visualizing the results

    + +

    + + +

    # visual representation of grid search
+# uses seaborn heatmap, could probably do this in matplotlib
+import seaborn as sns
+
+sns.set()
+
+train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
+test_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
+
+for i in range(len(eta_vals)):
+    for j in range(len(lmbd_vals)):
+        CNN = CNN_tf[i][j]
+
+        train_accuracy[i][j] = CNN.train_accuracy
+        test_accuracy[i][j] = CNN.test_accuracy
+
+        
+fig, ax = plt.subplots(figsize = (10, 10))
+sns.heatmap(train_accuracy, annot=True, ax=ax, cmap="viridis")
+ax.set_title("Training Accuracy")
+ax.set_ylabel("$\eta$")
+ax.set_xlabel("$\lambda$")
+plt.show()
+
+fig, ax = plt.subplots(figsize = (10, 10))
+sns.heatmap(test_accuracy, annot=True, ax=ax, cmap="viridis")
+ax.set_title("Test Accuracy")
+ax.set_ylabel("$\eta$")
+ax.set_xlabel("$\lambda$")
+plt.show()
+
    +
    + + +
    +

    Running with Keras

    + +

    + + +

    from keras.models import Sequential
+from keras.layers.convolutional import Conv2D
+from keras.layers.convolutional import MaxPooling2D
+from keras.layers import Flatten
+from keras.layers import Dense
+from keras.regularizers import l2
+from keras.optimizers import SGD
+
+def create_convolutional_neural_network_keras(input_shape, receptive_field,
+                                              n_filters, n_neurons_connected, n_categories,
+                                              eta, lmbd):
+    model = Sequential()
+    model.add(Conv2D(n_filters, (receptive_field, receptive_field), input_shape=input_shape, padding='same',
+              activation='relu', kernel_regularizer=l2(lmbd)))
+    model.add(MaxPooling2D(pool_size=(2, 2)))
+    model.add(Flatten())
+    model.add(Dense(n_neurons_connected, activation='relu', kernel_regularizer=l2(lmbd)))
+    model.add(Dense(n_categories, activation='softmax', kernel_regularizer=l2(lmbd)))
+    
+    sgd = SGD(lr=eta)
+    model.compile(loss='categorical_crossentropy', optimizer=sgd, metrics=['accuracy'])
+    
+    return model
+
+epochs = 100
+batch_size = 100
+input_shape = X_train.shape[1:4]
+receptive_field = 3
+n_filters = 10
+n_neurons_connected = 50
+n_categories = 10
+
+eta_vals = np.logspace(-5, 1, 7)
+lmbd_vals = np.logspace(-5, 1, 7)
+
    +
    + + +
    +

    Final part

    + +

    + + +

    CNN_keras = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)
+        
+for i, eta in enumerate(eta_vals):
+    for j, lmbd in enumerate(lmbd_vals):
+        CNN = create_convolutional_neural_network_keras(input_shape, receptive_field,
+                                              n_filters, n_neurons_connected, n_categories,
+                                              eta, lmbd)
+        CNN.fit(X_train, Y_train, epochs=epochs, batch_size=batch_size, verbose=0)
+        scores = CNN.evaluate(X_test, Y_test)
+        
+        CNN_keras[i][j] = CNN
+        
+        print("Learning rate = ", eta)
+        print("Lambda = ", lmbd)
+        print("Test accuracy: %.3f" % scores[1])
+        print()
+
    +
    + + +
    +

    Final visualization

    + +

    + + +

    # visual representation of grid search
+# uses seaborn heatmap, could probably do this in matplotlib
+import seaborn as sns
+
+sns.set()
+
+train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
+test_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
+
+for i in range(len(eta_vals)):
+    for j in range(len(lmbd_vals)):
+        CNN = CNN_keras[i][j]
+
+        train_accuracy[i][j] = CNN.evaluate(X_train, Y_train)[1]
+        test_accuracy[i][j] = CNN.evaluate(X_test, Y_test)[1]
+
+        
+fig, ax = plt.subplots(figsize = (10, 10))
+sns.heatmap(train_accuracy, annot=True, ax=ax, cmap="viridis")
+ax.set_title("Training Accuracy")
+ax.set_ylabel("$\eta$")
+ax.set_xlabel("$\lambda$")
+plt.show()
+
+fig, ax = plt.subplots(figsize = (10, 10))
+sns.heatmap(test_accuracy, annot=True, ax=ax, cmap="viridis")
+ax.set_title("Test Accuracy")
+ax.set_ylabel("$\eta$")
+ax.set_xlabel("$\lambda$")
+plt.show()
+
    +
    + + +
    +

    Fun links

    + +
      +

    1. Self-Driving cars using a convolutional neural network
      2. +

    2. Abstract art using convolutional neural networks
      3. +
    +
    + + + +
    +
    + + + + + + + + + + + + diff --git a/doc/src/CNN/cnn-solarized.html b/doc/src/CNN/cnn-solarized.html new file mode 100644 index 000000000..4979fb1bb --- /dev/null +++ b/doc/src/CNN/cnn-solarized.html @@ -0,0 +1,794 @@ + + + + + + + + +Convolutional Neural Networks + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

    Convolutional Neural Networks

    + +

    + + +

    +Morten Hjorth-Jensen [1, 2] +
    + +

    + + +

    [1] Department of Physics, University of Oslo
    +
    [2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University
    +
    +

    +

    Oct 3, 2019

    +
    +

    +









    + +

    Convolutional Neural Networks (recognizing images)

    + +

    +Convolutional Neural Networks (CNN) are very similar to ordinary Neural Networks. + +

    +They are made up of neurons that have learnable weights and +biases. Each neuron receives some inputs, performs a dot product and +optionally follows it with a non-linearity. The whole network still +expresses a single differentiable score function: from the raw image +pixels on one end to class scores at the other. And they still have a +loss function (for example Softmax) on the last (fully-connected) layer +and all the tips/tricks we developed for learning regular Neural +Networks still apply (back propagation, gradient descent etc etc). + +

    +What is the difference? CNN architectures make the explicit assumption that +the inputs are images, which allows us to encode certain properties +into the architecture. These then make the forward function more +efficient to implement and vastly reduce the amount of parameters in +the network. + +

    +Here we provide only a superficial overview, for the more interested, we recommend highly the course +IN5400 – Machine Learning for Image Analysis +and the slides of CS231. + +

    +









    + +

    Regular NNs don’t scale well to full images

    + +

    +As an example, consider +an image of size \( 32\times 32\times 3 \) (32 wide, 32 high, 3 color channels), so a +single fully-connected neuron in a first hidden layer of a regular +Neural Network would have \( 32\times 32\times 3 = 3072 \) weights. This amount still +seems manageable, but clearly this fully-connected structure does not +scale to larger images. For example, an image of more respectable +size, say \( 200\times 200\times 3 \), would lead to neurons that have +\( 200\times 200\times 3 = 120,000 \) weights. + +

    +We could have +several such neurons, and the parameters would add up quickly! Clearly, +this full connectivity is wasteful and the huge number of parameters +would quickly lead to possible overfitting. + +

    +

    +
    +

    Figure 1: A regular 3-layer Neural Network.

    +

    +
    + +

    +









    + +

    3D volumes of neurons

    + +

    +Convolutional Neural Networks take advantage of the fact that the +input consists of images and they constrain the architecture in a more +sensible way. + +

    +In particular, unlike a regular Neural Network, the +layers of a CNN have neurons arranged in 3 dimensions: width, +height, depth. (Note that the word depth here refers to the third +dimension of an activation volume, not to the depth of a full Neural +Network, which can refer to the total number of layers in a network.) + +

    +To understand it better, the above example of an image +with an input volume of +activations has dimensions \( 32\times 32\times 3 \) (width, height, +depth respectively). + +

    +The neurons in a layer will +only be connected to a small region of the layer before it, instead of +all of the neurons in a fully-connected manner. Moreover, the final +output layer could for this specific image have dimensions \( 1\times 1 \times 10 \), +because by the +end of the CNN architecture we will reduce the full image into a +single vector of class scores, arranged along the depth +dimension. + +

    +

    +
    +

    Figure 2: A CNN arranges its neurons in three dimensions (width, height, depth), as visualized in one of the layers. Every layer of a CNN transforms the 3D input volume to a 3D output volume of neuron activations. In this example, the red input layer holds the image, so its width and height would be the dimensions of the image, and the depth would be 3 (Red, Green, Blue channels).

    +

    +
    + +

    + + +

    Layers used to build CNNs

    + +

    +A simple CNN is a sequence of layers, and every layer of a CNN +transforms one volume of activations to another through a +differentiable function. We use three main types of layers to build +CNN architectures: Convolutional Layer, Pooling Layer, and +Fully-Connected Layer (exactly as seen in regular Neural Networks). We +will stack these layers to form a full CNN architecture. + +

    +A simple CNN for image classification could have the architecture: + +

      +
    • INPUT (\( 32\times 32 \times 3 \)) will hold the raw pixel values of the image, in this case an image of width 32, height 32, and with three color channels R,G,B.
      • +
    • CONV (convolutional )layer will compute the output of neurons that are connected to local regions in the input, each computing a dot product between their weights and a small region they are connected to in the input volume. This may result in volume such as \( [32\times 32\times 12] \) if we decided to use 12 filters.
      • +
    • RELU layer will apply an elementwise activation function, such as the \( max(0,x) \) thresholding at zero. This leaves the size of the volume unchanged (\( [32\times 32\times 12] \)).
      • +
    • POOL (pooling) layer will perform a downsampling operation along the spatial dimensions (width, height), resulting in volume such as \( [16\times 16\times 12] \).
      • +
    • FC (i.e. fully-connected) layer will compute the class scores, resulting in volume of size \( [1\times 1\times 10] \), where each of the 10 numbers correspond to a class score, such as among the 10 categories of the MNIST images we considered above . As with ordinary Neural Networks and as the name implies, each neuron in this layer will be connected to all the numbers in the previous volume.
      • +
    + +









    + +

    Transforming images

    + +

    +CNNs transform the original image layer by layer from the original +pixel values to the final class scores. + +

    +Observe that some layers contain +parameters and other don’t. In particular, the CNN layers perform +transformations that are a function of not only the activations in the +input volume, but also of the parameters (the weights and biases of +the neurons). On the other hand, the RELU/POOL layers will implement a +fixed function. The parameters in the CONV/FC layers will be trained +with gradient descent so that the class scores that the CNN computes +are consistent with the labels in the training set for each image. + +

    +









    + +

    CNNs in brief

    + +

    +In summary: + +

      +
    • A CNN architecture is in the simplest case a list of Layers that transform the image volume into an output volume (e.g. holding the class scores)
      • +
    • There are a few distinct types of Layers (e.g. CONV/FC/RELU/POOL are by far the most popular)
      • +
    • Each Layer accepts an input 3D volume and transforms it to an output 3D volume through a differentiable function
      • +
    • Each Layer may or may not have parameters (e.g. CONV/FC do, RELU/POOL don’t)
      • +
    • Each Layer may or may not have additional hyperparameters (e.g. CONV/FC/POOL do, RELU doesn’t)
      • +
    + +For more material on convolutional networks, we strongly recommend +the course +IN5400 – Machine Learning for Image Analysis +and the slides of CS231 which is taught at Stanford University (consistently ranked as one of the top computer science programs in the world). Michael Nielsen's book is a must read, in particular chapter 6 which deals with CNNs. + +

    +









    + +

    CNNs in more detail, building convolutional neural networks in Tensorflow and Keras

    + +

    +As discussed above, CNNs are neural networks built from the assumption that the inputs +to the network are 2D images. This is important because the number of features or pixels in images +grows very fast with the image size, and an enormous number of weights and biases are needed in order to build an accurate network. + +

    +As before, we still have our input, a hidden layer and an output. What's novel about convolutional networks +are the convolutional and pooling layers stacked in pairs between the input and the hidden layer. +In addition, the data is no longer represented as a 2D feature matrix, instead each input is a number of 2D +matrices, typically 1 for each color dimension (Red, Green, Blue). + +

    +









    + +

    Setting it up

    + +

    +It means that to represent the entire +dataset of images, we require a 4D matrix or tensor. This tensor has the dimensions: +$$ +(n_{inputs},\, n_{pixels, width},\, n_{pixels, height},\, depth) . +$$ + +

    +









    + +

    The MNIST dataset again

    + +

    +The MNIST dataset consists of grayscale images with a pixel size of +\( 28\times 28 \), meaning we require \( 28 \times 28 = 724 \) weights to each +neuron in the first hidden layer. + +

    +If we were to analyze images of size \( 128\times 128 \) we would require +\( 128 \times 128 = 16384 \) weights to each neuron. Even worse if we were +dealing with color images, as most images are, we have an image matrix +of size \( 128\times 128 \) for each color dimension (Red, Green, Blue), +meaning 3 times the number of weights \( = 49152 \) are required for every +single neuron in the first hidden layer. + +

    +









    + +

    Strong correlations

    +Images typically have strong local correlations, meaning that a small +part of the image varies little from its neighboring regions. If for +example we have an image of a blue car, we can roughly assume that a +small blue part of the image is surrounded by other blue regions. + +

    +Therefore, instead of connecting every single pixel to a neuron in the +first hidden layer, as we have previously done with deep neural +networks, we can instead connect each neuron to a small part of the +image (in all 3 RGB depth dimensions). The size of each small area is +fixed, and known as a receptive. + +

    + + +

    Layers of a CNN

    +The layers of a convolutional neural network arrange neurons in 3D: width, height and depth. +The input image is typically a square matrix of depth 3. + +

    +A convolution is performed on the image which outputs +a 3D volume of neurons. The weights to the input are arranged in a number of 2D matrices, known as filters. + +

    +Each filter slides along the input image, taking the dot product +between each small part of the image and the filter, in all depth +dimensions. This is then passed through a non-linear function, +typically the Rectified Linear (ReLu) function, which serves as the +activation of the neurons in the first convolutional layer. This is +further passed through a pooling layer, which reduces the size of the +convolutional layer, e.g. by taking the maximum or average across some +small regions, and this serves as input to the next convolutional +layer. + +

    +









    + +

    Systematic reduction

    + +

    +By systematically reducing the size of the input volume, through +convolution and pooling, the network should create representations of +small parts of the input, and then from them assemble representations +of larger areas. The final pooling layer is flattened to serve as +input to a hidden layer, such that each neuron in the final pooling +layer is connected to every single neuron in the hidden layer. This +then serves as input to the output layer, e.g. a softmax output for +classification. + +

    +









    + +

    Prerequisites: Collect and pre-process data

    +

    + + +

    # import necessary packages
+import numpy as np
+import matplotlib.pyplot as plt
+from sklearn import datasets
+
+
+# ensure the same random numbers appear every time
+np.random.seed(0)
+
+# display images in notebook
+%matplotlib inline
+plt.rcParams['figure.figsize'] = (12,12)
+
+
+# download MNIST dataset
+digits = datasets.load_digits()
+
+# define inputs and labels
+inputs = digits.images
+labels = digits.target
+
+# RGB images have a depth of 3
+# our images are grayscale so they should have a depth of 1
+inputs = inputs[:,:,:,np.newaxis]
+
+print("inputs = (n_inputs, pixel_width, pixel_height, depth) = " + str(inputs.shape))
+print("labels = (n_inputs) = " + str(labels.shape))
+
+
+# choose some random images to display
+n_inputs = len(inputs)
+indices = np.arange(n_inputs)
+random_indices = np.random.choice(indices, size=5)
+
+for i, image in enumerate(digits.images[random_indices]):
+    plt.subplot(1, 5, i+1)
+    plt.axis('off')
+    plt.imshow(image, cmap=plt.cm.gray_r, interpolation='nearest')
+    plt.title("Label: %d" % digits.target[random_indices[i]])
+plt.show()
+
    +

    +









    + +

    Importing Keras and Tensorflow

    +

    + + +

    from keras.utils import to_categorical
+from sklearn.model_selection import train_test_split
+
+# representation of labels
+labels = to_categorical(labels)
+
+# split into train and test data
+# one-liner from scikit-learn library
+train_size = 0.8
+test_size = 1 - train_size
+X_train, X_test, Y_train, Y_test = train_test_split(inputs, labels, train_size=train_size,
+                                                    test_size=test_size)
+
    +

    +









    + +

    Using TensorFlow backend

    + +

    +We need to define model and architecture and choose cost function and optmizer. +

    + + +

    import tensorflow as tf
+
+class ConvolutionalNeuralNetworkTensorflow:
+    def __init__(
+            self,
+            X_train,
+            Y_train,
+            X_test,
+            Y_test,
+            n_filters=10,
+            n_neurons_connected=50,
+            n_categories=10,
+            receptive_field=3,
+            stride=1,
+            padding=1,
+            epochs=10,
+            batch_size=100,
+            eta=0.1,
+            lmbd=0.0):
+        
+        self.global_step = tf.Variable(0, dtype=tf.int32, trainable=False, name='global_step')
+        
+        self.X_train = X_train
+        self.Y_train = Y_train
+        self.X_test = X_test
+        self.Y_test = Y_test
+        
+        self.n_inputs, self.input_width, self.input_height, self.depth = X_train.shape
+        
+        self.n_filters = n_filters
+        self.n_downsampled = int(self.input_width*self.input_height*n_filters / 4)
+        self.n_neurons_connected = n_neurons_connected
+        self.n_categories = n_categories
+        
+        self.receptive_field = receptive_field
+        self.stride = stride
+        self.strides = [stride, stride, stride, stride]
+        self.padding = padding
+        
+        self.epochs = epochs
+        self.batch_size = batch_size
+        self.iterations = self.n_inputs // self.batch_size
+        self.eta = eta
+        self.lmbd = lmbd
+        
+        self.create_placeholders()
+        self.create_CNN()
+        self.create_loss()
+        self.create_optimiser()
+        self.create_accuracy()
+    
+    def create_placeholders(self):
+        with tf.name_scope('data'):
+            self.X = tf.placeholder(tf.float32, shape=(None, self.input_width, self.input_height, self.depth), name='X_data')
+            self.Y = tf.placeholder(tf.float32, shape=(None, self.n_categories), name='Y_data')
+    
+    def create_CNN(self):
+        with tf.name_scope('CNN'):
+            
+            # Convolutional layer
+            self.W_conv = self.weight_variable([self.receptive_field, self.receptive_field, self.depth, self.n_filters], name='conv', dtype=tf.float32)
+            b_conv = self.weight_variable([self.n_filters], name='conv', dtype=tf.float32)
+            z_conv = tf.nn.conv2d(self.X, self.W_conv, self.strides, padding='SAME', name='conv') + b_conv
+            a_conv = tf.nn.relu(z_conv)
+            
+            # 2x2 max pooling
+            a_pool = tf.nn.max_pool(a_conv, [1, 2, 2, 1], [1, 2, 2, 1], padding='SAME', name='pool')
+            
+            # Fully connected layer
+            a_pool_flat = tf.reshape(a_pool, [-1, self.n_downsampled])
+            self.W_fc = self.weight_variable([self.n_downsampled, self.n_neurons_connected], name='fc', dtype=tf.float32)
+            b_fc = self.bias_variable([self.n_neurons_connected], name='fc', dtype=tf.float32)
+            a_fc = tf.nn.relu(tf.matmul(a_pool_flat, self.W_fc) + b_fc)
+            
+            # Output layer
+            self.W_out = self.weight_variable([self.n_neurons_connected, self.n_categories], name='out', dtype=tf.float32)
+            b_out = self.bias_variable([self.n_categories], name='out', dtype=tf.float32)
+            self.z_out = tf.matmul(a_fc, self.W_out) + b_out
+    
+    def create_loss(self):
+        with tf.name_scope('loss'):
+            softmax_loss = tf.reduce_mean(tf.nn.softmax_cross_entropy_with_logits_v2(labels=self.Y, logits=self.z_out))
+            
+            regularizer_loss_conv = tf.nn.l2_loss(self.W_conv)
+            regularizer_loss_fc = tf.nn.l2_loss(self.W_fc)
+            regularizer_loss_out = tf.nn.l2_loss(self.W_out)
+            regularizer_loss = self.lmbd*(regularizer_loss_conv + regularizer_loss_fc + regularizer_loss_out)
+            
+            self.loss = softmax_loss + regularizer_loss
+
+    def create_accuracy(self):
+        with tf.name_scope('accuracy'):
+            probabilities = tf.nn.softmax(self.z_out)
+            predictions = tf.argmax(probabilities, 1)
+            labels = tf.argmax(self.Y, 1)
+            
+            correct_predictions = tf.equal(predictions, labels)
+            correct_predictions = tf.cast(correct_predictions, tf.float32)
+            self.accuracy = tf.reduce_mean(correct_predictions)
+    
+    def create_optimiser(self):
+        with tf.name_scope('optimizer'):
+            self.optimizer = tf.train.GradientDescentOptimizer(learning_rate=self.eta).minimize(self.loss, global_step=self.global_step)
+            
+    def weight_variable(self, shape, name='', dtype=tf.float32):
+        initial = tf.truncated_normal(shape, stddev=0.1)
+        return tf.Variable(initial, name=name, dtype=dtype)
+    
+    def bias_variable(self, shape, name='', dtype=tf.float32):
+        initial = tf.constant(0.1, shape=shape)
+        return tf.Variable(initial, name=name, dtype=dtype)
+
+    def fit(self):
+        data_indices = np.arange(self.n_inputs)
+
+        with tf.Session() as sess:
+            sess.run(tf.global_variables_initializer())
+            for i in range(self.epochs):
+                for j in range(self.iterations):
+                    chosen_datapoints = np.random.choice(data_indices, size=self.batch_size, replace=False)
+                    batch_X, batch_Y = self.X_train[chosen_datapoints], self.Y_train[chosen_datapoints]
+            
+                    sess.run([CNN.loss, CNN.optimizer],
+                        feed_dict={CNN.X: batch_X,
+                                   CNN.Y: batch_Y})
+                    accuracy = sess.run(CNN.accuracy,
+                        feed_dict={CNN.X: batch_X,
+                                   CNN.Y: batch_Y})
+                    step = sess.run(CNN.global_step)
+    
+            self.train_loss, self.train_accuracy = sess.run([CNN.loss, CNN.accuracy],
+                feed_dict={CNN.X: self.X_train,
+                           CNN.Y: self.Y_train})
+        
+            self.test_loss, self.test_accuracy = sess.run([CNN.loss, CNN.accuracy],
+                feed_dict={CNN.X: self.X_test,
+                           CNN.Y: self.Y_test})
+
    +

    +









    + +

    Train the model

    + +

    +We need now to train the model, evaluate it and test its performance on test data, and eventually include hyperparameters. +

    + + +

    epochs = 100
+batch_size = 100
+n_filters = 10
+n_neurons_connected = 50
+n_categories = 10
+
+eta_vals = np.logspace(-5, 1, 7)
+lmbd_vals = np.logspace(-5, 1, 7)
+CNN_tf = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)
+        
+for i, eta in enumerate(eta_vals):
+    for j, lmbd in enumerate(lmbd_vals):
+        CNN = ConvolutionalNeuralNetworkTensorflow(X_train, Y_train, X_test, Y_test,
+                                      n_filters=n_filters, n_neurons_connected=n_neurons_connected,
+                                      n_categories=n_categories, epochs=epochs, batch_size=batch_size,
+                                      eta=eta, lmbd=lmbd)
+        CNN.fit()
+        
+        print("Learning rate = ", eta)
+        print("Lambda = ", lmbd)
+        print("Test accuracy: %.3f" % CNN.test_accuracy)
+        print()
+            
+        CNN_tf[i][j] = CNN
+
    +

    +









    + +

    Visualizing the results

    + +

    + + +

    # visual representation of grid search
+# uses seaborn heatmap, could probably do this in matplotlib
+import seaborn as sns
+
+sns.set()
+
+train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
+test_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
+
+for i in range(len(eta_vals)):
+    for j in range(len(lmbd_vals)):
+        CNN = CNN_tf[i][j]
+
+        train_accuracy[i][j] = CNN.train_accuracy
+        test_accuracy[i][j] = CNN.test_accuracy
+
+        
+fig, ax = plt.subplots(figsize = (10, 10))
+sns.heatmap(train_accuracy, annot=True, ax=ax, cmap="viridis")
+ax.set_title("Training Accuracy")
+ax.set_ylabel("$\eta$")
+ax.set_xlabel("$\lambda$")
+plt.show()
+
+fig, ax = plt.subplots(figsize = (10, 10))
+sns.heatmap(test_accuracy, annot=True, ax=ax, cmap="viridis")
+ax.set_title("Test Accuracy")
+ax.set_ylabel("$\eta$")
+ax.set_xlabel("$\lambda$")
+plt.show()
+
    +

    + + +

    Running with Keras

    + +

    + + +

    from keras.models import Sequential
+from keras.layers.convolutional import Conv2D
+from keras.layers.convolutional import MaxPooling2D
+from keras.layers import Flatten
+from keras.layers import Dense
+from keras.regularizers import l2
+from keras.optimizers import SGD
+
+def create_convolutional_neural_network_keras(input_shape, receptive_field,
+                                              n_filters, n_neurons_connected, n_categories,
+                                              eta, lmbd):
+    model = Sequential()
+    model.add(Conv2D(n_filters, (receptive_field, receptive_field), input_shape=input_shape, padding='same',
+              activation='relu', kernel_regularizer=l2(lmbd)))
+    model.add(MaxPooling2D(pool_size=(2, 2)))
+    model.add(Flatten())
+    model.add(Dense(n_neurons_connected, activation='relu', kernel_regularizer=l2(lmbd)))
+    model.add(Dense(n_categories, activation='softmax', kernel_regularizer=l2(lmbd)))
+    
+    sgd = SGD(lr=eta)
+    model.compile(loss='categorical_crossentropy', optimizer=sgd, metrics=['accuracy'])
+    
+    return model
+
+epochs = 100
+batch_size = 100
+input_shape = X_train.shape[1:4]
+receptive_field = 3
+n_filters = 10
+n_neurons_connected = 50
+n_categories = 10
+
+eta_vals = np.logspace(-5, 1, 7)
+lmbd_vals = np.logspace(-5, 1, 7)
+
    +

    +









    + +

    Final part

    + +

    + + +

    CNN_keras = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)
+        
+for i, eta in enumerate(eta_vals):
+    for j, lmbd in enumerate(lmbd_vals):
+        CNN = create_convolutional_neural_network_keras(input_shape, receptive_field,
+                                              n_filters, n_neurons_connected, n_categories,
+                                              eta, lmbd)
+        CNN.fit(X_train, Y_train, epochs=epochs, batch_size=batch_size, verbose=0)
+        scores = CNN.evaluate(X_test, Y_test)
+        
+        CNN_keras[i][j] = CNN
+        
+        print("Learning rate = ", eta)
+        print("Lambda = ", lmbd)
+        print("Test accuracy: %.3f" % scores[1])
+        print()
+
    +

    +









    + +

    Final visualization

    + +

    + + +

    # visual representation of grid search
+# uses seaborn heatmap, could probably do this in matplotlib
+import seaborn as sns
+
+sns.set()
+
+train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
+test_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
+
+for i in range(len(eta_vals)):
+    for j in range(len(lmbd_vals)):
+        CNN = CNN_keras[i][j]
+
+        train_accuracy[i][j] = CNN.evaluate(X_train, Y_train)[1]
+        test_accuracy[i][j] = CNN.evaluate(X_test, Y_test)[1]
+
+        
+fig, ax = plt.subplots(figsize = (10, 10))
+sns.heatmap(train_accuracy, annot=True, ax=ax, cmap="viridis")
+ax.set_title("Training Accuracy")
+ax.set_ylabel("$\eta$")
+ax.set_xlabel("$\lambda$")
+plt.show()
+
+fig, ax = plt.subplots(figsize = (10, 10))
+sns.heatmap(test_accuracy, annot=True, ax=ax, cmap="viridis")
+ax.set_title("Test Accuracy")
+ax.set_ylabel("$\eta$")
+ax.set_xlabel("$\lambda$")
+plt.show()
+
    +

    +









    + +

    Fun links

    + +
      +
    1. Self-Driving cars using a convolutional neural network
      2. +
    2. Abstract art using convolutional neural networks
      3. +
    + + + + + +
    + © 1999-2019, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license +
    + + + + + + diff --git a/doc/src/CNN/cnn.do.txt b/doc/src/CNN/cnn.do.txt new file mode 100644 index 000000000..e20014726 --- /dev/null +++ b/doc/src/CNN/cnn.do.txt @@ -0,0 +1,600 @@ +TITLE: Convolutional Neural Networks +AUTHOR: Morten Hjorth-Jensen {copyright, 1999-present|CC BY-NC} at Department of Physics, University of Oslo & Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University +DATE: today + + + +!split +===== Convolutional Neural Networks (recognizing images) ===== + +Convolutional Neural Networks (CNN) are very similar to ordinary Neural Networks. + +They are made up of neurons that have learnable weights and +biases. Each neuron receives some inputs, performs a dot product and +optionally follows it with a non-linearity. The whole network still +expresses a single differentiable score function: from the raw image +pixels on one end to class scores at the other. And they still have a +loss function (for example Softmax) on the last (fully-connected) layer +and all the tips/tricks we developed for learning regular Neural +Networks still apply (back propagation, gradient descent etc etc). + +What is the difference? _CNN architectures make the explicit assumption that +the inputs are images, which allows us to encode certain properties +into the architecture. These then make the forward function more +efficient to implement and vastly reduce the amount of parameters in +the network._ + +Here we provide only a superficial overview, for the more interested, we recommend highly the course +"IN5400 – Machine Learning for Image Analysis":"https://www.uio.no/studier/emner/matnat/ifi/IN5400/index-eng.html" +and the slides of "CS231":"http://cs231n.github.io/convolutional-networks/". + +!split +===== Regular NNs don’t scale well to full images ===== + +As an example, consider +an image of size $32\times 32\times 3$ (32 wide, 32 high, 3 color channels), so a +single fully-connected neuron in a first hidden layer of a regular +Neural Network would have $32\times 32\times 3 = 3072$ weights. This amount still +seems manageable, but clearly this fully-connected structure does not +scale to larger images. For example, an image of more respectable +size, say $200\times 200\times 3$, would lead to neurons that have +$200\times 200\times 3 = 120,000$ weights. + +We could have +several such neurons, and the parameters would add up quickly! Clearly, +this full connectivity is wasteful and the huge number of parameters +would quickly lead to possible overfitting. + +FIGURE: [figslides/nn.jpeg, width=500 frac=0.6] A regular 3-layer Neural Network. + +!split +===== 3D volumes of neurons ===== + +Convolutional Neural Networks take advantage of the fact that the +input consists of images and they constrain the architecture in a more +sensible way. + +In particular, unlike a regular Neural Network, the +layers of a CNN have neurons arranged in 3 dimensions: width, +height, depth. (Note that the word depth here refers to the third +dimension of an activation volume, not to the depth of a full Neural +Network, which can refer to the total number of layers in a network.) + +To understand it better, the above example of an image +with an input volume of +activations has dimensions $32\times 32\times 3$ (width, height, +depth respectively). + +The neurons in a layer will +only be connected to a small region of the layer before it, instead of +all of the neurons in a fully-connected manner. Moreover, the final +output layer could for this specific image have dimensions $1\times 1 \times 10$, +because by the +end of the CNN architecture we will reduce the full image into a +single vector of class scores, arranged along the depth +dimension. + +FIGURE: [figslides/cnn.jpeg, width=500 frac=0.6] A CNN arranges its neurons in three dimensions (width, height, depth), as visualized in one of the layers. Every layer of a CNN transforms the 3D input volume to a 3D output volume of neuron activations. In this example, the red input layer holds the image, so its width and height would be the dimensions of the image, and the depth would be 3 (Red, Green, Blue channels). + + + +!split +===== Layers used to build CNNs ===== + + +A simple CNN is a sequence of layers, and every layer of a CNN +transforms one volume of activations to another through a +differentiable function. We use three main types of layers to build +CNN architectures: Convolutional Layer, Pooling Layer, and +Fully-Connected Layer (exactly as seen in regular Neural Networks). We +will stack these layers to form a full CNN architecture. + +A simple CNN for image classification could have the architecture: + +* _INPUT_ ($32\times 32 \times 3$) will hold the raw pixel values of the image, in this case an image of width 32, height 32, and with three color channels R,G,B. +* _CONV_ (convolutional )layer will compute the output of neurons that are connected to local regions in the input, each computing a dot product between their weights and a small region they are connected to in the input volume. This may result in volume such as $[32\times 32\times 12]$ if we decided to use 12 filters. +* _RELU_ layer will apply an elementwise activation function, such as the $max(0,x)$ thresholding at zero. This leaves the size of the volume unchanged ($[32\times 32\times 12]$). +* _POOL_ (pooling) layer will perform a downsampling operation along the spatial dimensions (width, height), resulting in volume such as $[16\times 16\times 12]$. +* _FC_ (i.e. fully-connected) layer will compute the class scores, resulting in volume of size $[1\times 1\times 10]$, where each of the 10 numbers correspond to a class score, such as among the 10 categories of the MNIST images we considered above . As with ordinary Neural Networks and as the name implies, each neuron in this layer will be connected to all the numbers in the previous volume. + + +!split +===== Transforming images ===== + +CNNs transform the original image layer by layer from the original +pixel values to the final class scores. + +Observe that some layers contain +parameters and other don’t. In particular, the CNN layers perform +transformations that are a function of not only the activations in the +input volume, but also of the parameters (the weights and biases of +the neurons). On the other hand, the RELU/POOL layers will implement a +fixed function. The parameters in the CONV/FC layers will be trained +with gradient descent so that the class scores that the CNN computes +are consistent with the labels in the training set for each image. + + +!split +===== CNNs in brief ===== + +In summary: + +* A CNN architecture is in the simplest case a list of Layers that transform the image volume into an output volume (e.g. holding the class scores) +* There are a few distinct types of Layers (e.g. CONV/FC/RELU/POOL are by far the most popular) +* Each Layer accepts an input 3D volume and transforms it to an output 3D volume through a differentiable function +* Each Layer may or may not have parameters (e.g. CONV/FC do, RELU/POOL don’t) +* Each Layer may or may not have additional hyperparameters (e.g. CONV/FC/POOL do, RELU doesn’t) + +For more material on convolutional networks, we strongly recommend +the course +"IN5400 – Machine Learning for Image Analysis":"https://www.uio.no/studier/emner/matnat/ifi/IN5400/index-eng.html" +and the slides of "CS231":"http://cs231n.github.io/convolutional-networks/" which is taught at Stanford University (consistently ranked as one of the top computer science programs in the world). "Michael Nielsen's book is a must read, in particular chapter 6 which deals with CNNs":"http://neuralnetworksanddeeplearning.com/chap6.html". + + +!split +===== CNNs in more detail, building convolutional neural networks in Tensorflow and Keras ===== + + +As discussed above, CNNs are neural networks built from the assumption that the inputs +to the network are 2D images. This is important because the number of features or pixels in images +grows very fast with the image size, and an enormous number of weights and biases are needed in order to build an accurate network. + +As before, we still have our input, a hidden layer and an output. What's novel about convolutional networks +are the _convolutional_ and _pooling_ layers stacked in pairs between the input and the hidden layer. +In addition, the data is no longer represented as a 2D feature matrix, instead each input is a number of 2D +matrices, typically 1 for each color dimension (Red, Green, Blue). + + +!split +===== Setting it up ===== + +It means that to represent the entire +dataset of images, we require a 4D matrix or _tensor_. This tensor has the dimensions: +!bt +\[ +(n_{inputs},\, n_{pixels, width},\, n_{pixels, height},\, depth) . +\] +!et + +!split +===== The MNIST dataset again ===== + +The MNIST dataset consists of grayscale images with a pixel size of +$28\times 28$, meaning we require $28 \times 28 = 724$ weights to each +neuron in the first hidden layer. + +If we were to analyze images of size $128\times 128$ we would require +$128 \times 128 = 16384$ weights to each neuron. Even worse if we were +dealing with color images, as most images are, we have an image matrix +of size $128\times 128$ for each color dimension (Red, Green, Blue), +meaning 3 times the number of weights $= 49152$ are required for every +single neuron in the first hidden layer. + + +!split +===== Strong correlations ===== +Images typically have strong local correlations, meaning that a small +part of the image varies little from its neighboring regions. If for +example we have an image of a blue car, we can roughly assume that a +small blue part of the image is surrounded by other blue regions. + +Therefore, instead of connecting every single pixel to a neuron in the +first hidden layer, as we have previously done with deep neural +networks, we can instead connect each neuron to a small part of the +image (in all 3 RGB depth dimensions). The size of each small area is +fixed, and known as a "receptive":"https://en.wikipedia.org/wiki/Receptive_field". + + +!split +===== Layers of a CNN ===== +The layers of a convolutional neural network arrange neurons in 3D: width, height and depth. +The input image is typically a square matrix of depth 3. + +A _convolution_ is performed on the image which outputs +a 3D volume of neurons. The weights to the input are arranged in a number of 2D matrices, known as _filters_. + + +Each filter slides along the input image, taking the dot product +between each small part of the image and the filter, in all depth +dimensions. This is then passed through a non-linear function, +typically the _Rectified Linear (ReLu)_ function, which serves as the +activation of the neurons in the first convolutional layer. This is +further passed through a _pooling layer_, which reduces the size of the +convolutional layer, e.g. by taking the maximum or average across some +small regions, and this serves as input to the next convolutional +layer. + + +!split +===== Systematic reduction ===== + +By systematically reducing the size of the input volume, through +convolution and pooling, the network should create representations of +small parts of the input, and then from them assemble representations +of larger areas. The final pooling layer is flattened to serve as +input to a hidden layer, such that each neuron in the final pooling +layer is connected to every single neuron in the hidden layer. This +then serves as input to the output layer, e.g. a softmax output for +classification. + + +!split +===== Prerequisites: Collect and pre-process data ===== +!bc pycod +# import necessary packages +import numpy as np +import matplotlib.pyplot as plt +from sklearn import datasets + + +# ensure the same random numbers appear every time +np.random.seed(0) + +# display images in notebook +%matplotlib inline +plt.rcParams['figure.figsize'] = (12,12) + + +# download MNIST dataset +digits = datasets.load_digits() + +# define inputs and labels +inputs = digits.images +labels = digits.target + +# RGB images have a depth of 3 +# our images are grayscale so they should have a depth of 1 +inputs = inputs[:,:,:,np.newaxis] + +print("inputs = (n_inputs, pixel_width, pixel_height, depth) = " + str(inputs.shape)) +print("labels = (n_inputs) = " + str(labels.shape)) + + +# choose some random images to display +n_inputs = len(inputs) +indices = np.arange(n_inputs) +random_indices = np.random.choice(indices, size=5) + +for i, image in enumerate(digits.images[random_indices]): + plt.subplot(1, 5, i+1) + plt.axis('off') + plt.imshow(image, cmap=plt.cm.gray_r, interpolation='nearest') + plt.title("Label: %d" % digits.target[random_indices[i]]) +plt.show() +!ec + + +!split +===== Importing Keras and Tensorflow ===== +!bc pycod +from keras.utils import to_categorical +from sklearn.model_selection import train_test_split + +# representation of labels +labels = to_categorical(labels) + +# split into train and test data +# one-liner from scikit-learn library +train_size = 0.8 +test_size = 1 - train_size +X_train, X_test, Y_train, Y_test = train_test_split(inputs, labels, train_size=train_size, + test_size=test_size) +!ec + +!split +===== Using TensorFlow backend ===== + +We need to define model and architecture and choose cost function and optmizer. +!bc pycid + +import tensorflow as tf + +class ConvolutionalNeuralNetworkTensorflow: + def __init__( + self, + X_train, + Y_train, + X_test, + Y_test, + n_filters=10, + n_neurons_connected=50, + n_categories=10, + receptive_field=3, + stride=1, + padding=1, + epochs=10, + batch_size=100, + eta=0.1, + lmbd=0.0): + + self.global_step = tf.Variable(0, dtype=tf.int32, trainable=False, name='global_step') + + self.X_train = X_train + self.Y_train = Y_train + self.X_test = X_test + self.Y_test = Y_test + + self.n_inputs, self.input_width, self.input_height, self.depth = X_train.shape + + self.n_filters = n_filters + self.n_downsampled = int(self.input_width*self.input_height*n_filters / 4) + self.n_neurons_connected = n_neurons_connected + self.n_categories = n_categories + + self.receptive_field = receptive_field + self.stride = stride + self.strides = [stride, stride, stride, stride] + self.padding = padding + + self.epochs = epochs + self.batch_size = batch_size + self.iterations = self.n_inputs // self.batch_size + self.eta = eta + self.lmbd = lmbd + + self.create_placeholders() + self.create_CNN() + self.create_loss() + self.create_optimiser() + self.create_accuracy() + + def create_placeholders(self): + with tf.name_scope('data'): + self.X = tf.placeholder(tf.float32, shape=(None, self.input_width, self.input_height, self.depth), name='X_data') + self.Y = tf.placeholder(tf.float32, shape=(None, self.n_categories), name='Y_data') + + def create_CNN(self): + with tf.name_scope('CNN'): + + # Convolutional layer + self.W_conv = self.weight_variable([self.receptive_field, self.receptive_field, self.depth, self.n_filters], name='conv', dtype=tf.float32) + b_conv = self.weight_variable([self.n_filters], name='conv', dtype=tf.float32) + z_conv = tf.nn.conv2d(self.X, self.W_conv, self.strides, padding='SAME', name='conv') + b_conv + a_conv = tf.nn.relu(z_conv) + + # 2x2 max pooling + a_pool = tf.nn.max_pool(a_conv, [1, 2, 2, 1], [1, 2, 2, 1], padding='SAME', name='pool') + + # Fully connected layer + a_pool_flat = tf.reshape(a_pool, [-1, self.n_downsampled]) + self.W_fc = self.weight_variable([self.n_downsampled, self.n_neurons_connected], name='fc', dtype=tf.float32) + b_fc = self.bias_variable([self.n_neurons_connected], name='fc', dtype=tf.float32) + a_fc = tf.nn.relu(tf.matmul(a_pool_flat, self.W_fc) + b_fc) + + # Output layer + self.W_out = self.weight_variable([self.n_neurons_connected, self.n_categories], name='out', dtype=tf.float32) + b_out = self.bias_variable([self.n_categories], name='out', dtype=tf.float32) + self.z_out = tf.matmul(a_fc, self.W_out) + b_out + + def create_loss(self): + with tf.name_scope('loss'): + softmax_loss = tf.reduce_mean(tf.nn.softmax_cross_entropy_with_logits_v2(labels=self.Y, logits=self.z_out)) + + regularizer_loss_conv = tf.nn.l2_loss(self.W_conv) + regularizer_loss_fc = tf.nn.l2_loss(self.W_fc) + regularizer_loss_out = tf.nn.l2_loss(self.W_out) + regularizer_loss = self.lmbd*(regularizer_loss_conv + regularizer_loss_fc + regularizer_loss_out) + + self.loss = softmax_loss + regularizer_loss + + def create_accuracy(self): + with tf.name_scope('accuracy'): + probabilities = tf.nn.softmax(self.z_out) + predictions = tf.argmax(probabilities, 1) + labels = tf.argmax(self.Y, 1) + + correct_predictions = tf.equal(predictions, labels) + correct_predictions = tf.cast(correct_predictions, tf.float32) + self.accuracy = tf.reduce_mean(correct_predictions) + + def create_optimiser(self): + with tf.name_scope('optimizer'): + self.optimizer = tf.train.GradientDescentOptimizer(learning_rate=self.eta).minimize(self.loss, global_step=self.global_step) + + def weight_variable(self, shape, name='', dtype=tf.float32): + initial = tf.truncated_normal(shape, stddev=0.1) + return tf.Variable(initial, name=name, dtype=dtype) + + def bias_variable(self, shape, name='', dtype=tf.float32): + initial = tf.constant(0.1, shape=shape) + return tf.Variable(initial, name=name, dtype=dtype) + + def fit(self): + data_indices = np.arange(self.n_inputs) + + with tf.Session() as sess: + sess.run(tf.global_variables_initializer()) + for i in range(self.epochs): + for j in range(self.iterations): + chosen_datapoints = np.random.choice(data_indices, size=self.batch_size, replace=False) + batch_X, batch_Y = self.X_train[chosen_datapoints], self.Y_train[chosen_datapoints] + + sess.run([CNN.loss, CNN.optimizer], + feed_dict={CNN.X: batch_X, + CNN.Y: batch_Y}) + accuracy = sess.run(CNN.accuracy, + feed_dict={CNN.X: batch_X, + CNN.Y: batch_Y}) + step = sess.run(CNN.global_step) + + self.train_loss, self.train_accuracy = sess.run([CNN.loss, CNN.accuracy], + feed_dict={CNN.X: self.X_train, + CNN.Y: self.Y_train}) + + self.test_loss, self.test_accuracy = sess.run([CNN.loss, CNN.accuracy], + feed_dict={CNN.X: self.X_test, + CNN.Y: self.Y_test}) +!ec + +!split +===== Train the model ===== + +We need now to train the model, evaluate it and test its performance on test data, and eventually include hyperparameters. +!bc pycod +epochs = 100 +batch_size = 100 +n_filters = 10 +n_neurons_connected = 50 +n_categories = 10 + +eta_vals = np.logspace(-5, 1, 7) +lmbd_vals = np.logspace(-5, 1, 7) +CNN_tf = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object) + +for i, eta in enumerate(eta_vals): + for j, lmbd in enumerate(lmbd_vals): + CNN = ConvolutionalNeuralNetworkTensorflow(X_train, Y_train, X_test, Y_test, + n_filters=n_filters, n_neurons_connected=n_neurons_connected, + n_categories=n_categories, epochs=epochs, batch_size=batch_size, + eta=eta, lmbd=lmbd) + CNN.fit() + + print("Learning rate = ", eta) + print("Lambda = ", lmbd) + print("Test accuracy: %.3f" % CNN.test_accuracy) + print() + + CNN_tf[i][j] = CNN +!ec + +!split +===== Visualizing the results ===== + +!bc pycod +# visual representation of grid search +# uses seaborn heatmap, could probably do this in matplotlib +import seaborn as sns + +sns.set() + +train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals))) +test_accuracy = np.zeros((len(eta_vals), len(lmbd_vals))) + +for i in range(len(eta_vals)): + for j in range(len(lmbd_vals)): + CNN = CNN_tf[i][j] + + train_accuracy[i][j] = CNN.train_accuracy + test_accuracy[i][j] = CNN.test_accuracy + + +fig, ax = plt.subplots(figsize = (10, 10)) +sns.heatmap(train_accuracy, annot=True, ax=ax, cmap="viridis") +ax.set_title("Training Accuracy") +ax.set_ylabel("$\eta$") +ax.set_xlabel("$\lambda$") +plt.show() + +fig, ax = plt.subplots(figsize = (10, 10)) +sns.heatmap(test_accuracy, annot=True, ax=ax, cmap="viridis") +ax.set_title("Test Accuracy") +ax.set_ylabel("$\eta$") +ax.set_xlabel("$\lambda$") +plt.show() +!ec + +!split +===== Running with Keras ===== + +!bc pycod +from keras.models import Sequential +from keras.layers.convolutional import Conv2D +from keras.layers.convolutional import MaxPooling2D +from keras.layers import Flatten +from keras.layers import Dense +from keras.regularizers import l2 +from keras.optimizers import SGD + +def create_convolutional_neural_network_keras(input_shape, receptive_field, + n_filters, n_neurons_connected, n_categories, + eta, lmbd): + model = Sequential() + model.add(Conv2D(n_filters, (receptive_field, receptive_field), input_shape=input_shape, padding='same', + activation='relu', kernel_regularizer=l2(lmbd))) + model.add(MaxPooling2D(pool_size=(2, 2))) + model.add(Flatten()) + model.add(Dense(n_neurons_connected, activation='relu', kernel_regularizer=l2(lmbd))) + model.add(Dense(n_categories, activation='softmax', kernel_regularizer=l2(lmbd))) + + sgd = SGD(lr=eta) + model.compile(loss='categorical_crossentropy', optimizer=sgd, metrics=['accuracy']) + + return model + +epochs = 100 +batch_size = 100 +input_shape = X_train.shape[1:4] +receptive_field = 3 +n_filters = 10 +n_neurons_connected = 50 +n_categories = 10 + +eta_vals = np.logspace(-5, 1, 7) +lmbd_vals = np.logspace(-5, 1, 7) +!ec + +!split +===== Final part ===== + +!bc pycod +CNN_keras = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object) + +for i, eta in enumerate(eta_vals): + for j, lmbd in enumerate(lmbd_vals): + CNN = create_convolutional_neural_network_keras(input_shape, receptive_field, + n_filters, n_neurons_connected, n_categories, + eta, lmbd) + CNN.fit(X_train, Y_train, epochs=epochs, batch_size=batch_size, verbose=0) + scores = CNN.evaluate(X_test, Y_test) + + CNN_keras[i][j] = CNN + + print("Learning rate = ", eta) + print("Lambda = ", lmbd) + print("Test accuracy: %.3f" % scores[1]) + print() +!ec + +!split +===== Final visualization ===== + +!bc +# visual representation of grid search +# uses seaborn heatmap, could probably do this in matplotlib +import seaborn as sns + +sns.set() + +train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals))) +test_accuracy = np.zeros((len(eta_vals), len(lmbd_vals))) + +for i in range(len(eta_vals)): + for j in range(len(lmbd_vals)): + CNN = CNN_keras[i][j] + + train_accuracy[i][j] = CNN.evaluate(X_train, Y_train)[1] + test_accuracy[i][j] = CNN.evaluate(X_test, Y_test)[1] + + +fig, ax = plt.subplots(figsize = (10, 10)) +sns.heatmap(train_accuracy, annot=True, ax=ax, cmap="viridis") +ax.set_title("Training Accuracy") +ax.set_ylabel("$\eta$") +ax.set_xlabel("$\lambda$") +plt.show() + +fig, ax = plt.subplots(figsize = (10, 10)) +sns.heatmap(test_accuracy, annot=True, ax=ax, cmap="viridis") +ax.set_title("Test Accuracy") +ax.set_ylabel("$\eta$") +ax.set_xlabel("$\lambda$") +plt.show() +!ec + +!split +===== Fun links ===== + +o "Self-Driving cars using a convolutional neural network":"https://arxiv.org/abs/1604.07316" +o "Abstract art using convolutional neural networks":"https://deepdreamgenerator.com/" + + diff --git a/doc/src/CNN/cnn.html b/doc/src/CNN/cnn.html new file mode 100644 index 000000000..176bc3958 --- /dev/null +++ b/doc/src/CNN/cnn.html @@ -0,0 +1,799 @@ + + + + + + + + +Convolutional Neural Networks + + + + + + + + + + + + + + + + + + + + + + + +

    Convolutional Neural Networks

    + +

    + + +

    +Morten Hjorth-Jensen [1, 2] +
    + +

    + + +

    [1] Department of Physics, University of Oslo
    +
    [2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University
    +
    +

    +

    Oct 3, 2019

    +
    +

    +









    + +

    Convolutional Neural Networks (recognizing images)

    + +

    +Convolutional Neural Networks (CNN) are very similar to ordinary Neural Networks. + +

    +They are made up of neurons that have learnable weights and +biases. Each neuron receives some inputs, performs a dot product and +optionally follows it with a non-linearity. The whole network still +expresses a single differentiable score function: from the raw image +pixels on one end to class scores at the other. And they still have a +loss function (for example Softmax) on the last (fully-connected) layer +and all the tips/tricks we developed for learning regular Neural +Networks still apply (back propagation, gradient descent etc etc). + +

    +What is the difference? CNN architectures make the explicit assumption that +the inputs are images, which allows us to encode certain properties +into the architecture. These then make the forward function more +efficient to implement and vastly reduce the amount of parameters in +the network. + +

    +Here we provide only a superficial overview, for the more interested, we recommend highly the course +IN5400 – Machine Learning for Image Analysis +and the slides of CS231. + +

    +









    + +

    Regular NNs don’t scale well to full images

    + +

    +As an example, consider +an image of size \( 32\times 32\times 3 \) (32 wide, 32 high, 3 color channels), so a +single fully-connected neuron in a first hidden layer of a regular +Neural Network would have \( 32\times 32\times 3 = 3072 \) weights. This amount still +seems manageable, but clearly this fully-connected structure does not +scale to larger images. For example, an image of more respectable +size, say \( 200\times 200\times 3 \), would lead to neurons that have +\( 200\times 200\times 3 = 120,000 \) weights. + +

    +We could have +several such neurons, and the parameters would add up quickly! Clearly, +this full connectivity is wasteful and the huge number of parameters +would quickly lead to possible overfitting. + +

    +

    +
    +

    Figure 1: A regular 3-layer Neural Network.

    +

    +
    + +

    +









    + +

    3D volumes of neurons

    + +

    +Convolutional Neural Networks take advantage of the fact that the +input consists of images and they constrain the architecture in a more +sensible way. + +

    +In particular, unlike a regular Neural Network, the +layers of a CNN have neurons arranged in 3 dimensions: width, +height, depth. (Note that the word depth here refers to the third +dimension of an activation volume, not to the depth of a full Neural +Network, which can refer to the total number of layers in a network.) + +

    +To understand it better, the above example of an image +with an input volume of +activations has dimensions \( 32\times 32\times 3 \) (width, height, +depth respectively). + +

    +The neurons in a layer will +only be connected to a small region of the layer before it, instead of +all of the neurons in a fully-connected manner. Moreover, the final +output layer could for this specific image have dimensions \( 1\times 1 \times 10 \), +because by the +end of the CNN architecture we will reduce the full image into a +single vector of class scores, arranged along the depth +dimension. + +

    +

    +
    +

    Figure 2: A CNN arranges its neurons in three dimensions (width, height, depth), as visualized in one of the layers. Every layer of a CNN transforms the 3D input volume to a 3D output volume of neuron activations. In this example, the red input layer holds the image, so its width and height would be the dimensions of the image, and the depth would be 3 (Red, Green, Blue channels).

    +

    +
    + +

    + + +

    Layers used to build CNNs

    + +

    +A simple CNN is a sequence of layers, and every layer of a CNN +transforms one volume of activations to another through a +differentiable function. We use three main types of layers to build +CNN architectures: Convolutional Layer, Pooling Layer, and +Fully-Connected Layer (exactly as seen in regular Neural Networks). We +will stack these layers to form a full CNN architecture. + +

    +A simple CNN for image classification could have the architecture: + +

      +
    • INPUT (\( 32\times 32 \times 3 \)) will hold the raw pixel values of the image, in this case an image of width 32, height 32, and with three color channels R,G,B.
      • +
    • CONV (convolutional )layer will compute the output of neurons that are connected to local regions in the input, each computing a dot product between their weights and a small region they are connected to in the input volume. This may result in volume such as \( [32\times 32\times 12] \) if we decided to use 12 filters.
      • +
    • RELU layer will apply an elementwise activation function, such as the \( max(0,x) \) thresholding at zero. This leaves the size of the volume unchanged (\( [32\times 32\times 12] \)).
      • +
    • POOL (pooling) layer will perform a downsampling operation along the spatial dimensions (width, height), resulting in volume such as \( [16\times 16\times 12] \).
      • +
    • FC (i.e. fully-connected) layer will compute the class scores, resulting in volume of size \( [1\times 1\times 10] \), where each of the 10 numbers correspond to a class score, such as among the 10 categories of the MNIST images we considered above . As with ordinary Neural Networks and as the name implies, each neuron in this layer will be connected to all the numbers in the previous volume.
      • +
    + +









    + +

    Transforming images

    + +

    +CNNs transform the original image layer by layer from the original +pixel values to the final class scores. + +

    +Observe that some layers contain +parameters and other don’t. In particular, the CNN layers perform +transformations that are a function of not only the activations in the +input volume, but also of the parameters (the weights and biases of +the neurons). On the other hand, the RELU/POOL layers will implement a +fixed function. The parameters in the CONV/FC layers will be trained +with gradient descent so that the class scores that the CNN computes +are consistent with the labels in the training set for each image. + +

    +









    + +

    CNNs in brief

    + +

    +In summary: + +

      +
    • A CNN architecture is in the simplest case a list of Layers that transform the image volume into an output volume (e.g. holding the class scores)
      • +
    • There are a few distinct types of Layers (e.g. CONV/FC/RELU/POOL are by far the most popular)
      • +
    • Each Layer accepts an input 3D volume and transforms it to an output 3D volume through a differentiable function
      • +
    • Each Layer may or may not have parameters (e.g. CONV/FC do, RELU/POOL don’t)
      • +
    • Each Layer may or may not have additional hyperparameters (e.g. CONV/FC/POOL do, RELU doesn’t)
      • +
    + +For more material on convolutional networks, we strongly recommend +the course +IN5400 – Machine Learning for Image Analysis +and the slides of CS231 which is taught at Stanford University (consistently ranked as one of the top computer science programs in the world). Michael Nielsen's book is a must read, in particular chapter 6 which deals with CNNs. + +

    +









    + +

    CNNs in more detail, building convolutional neural networks in Tensorflow and Keras

    + +

    +As discussed above, CNNs are neural networks built from the assumption that the inputs +to the network are 2D images. This is important because the number of features or pixels in images +grows very fast with the image size, and an enormous number of weights and biases are needed in order to build an accurate network. + +

    +As before, we still have our input, a hidden layer and an output. What's novel about convolutional networks +are the convolutional and pooling layers stacked in pairs between the input and the hidden layer. +In addition, the data is no longer represented as a 2D feature matrix, instead each input is a number of 2D +matrices, typically 1 for each color dimension (Red, Green, Blue). + +

    +









    + +

    Setting it up

    + +

    +It means that to represent the entire +dataset of images, we require a 4D matrix or tensor. This tensor has the dimensions: +$$ +(n_{inputs},\, n_{pixels, width},\, n_{pixels, height},\, depth) . +$$ + +

    +









    + +

    The MNIST dataset again

    + +

    +The MNIST dataset consists of grayscale images with a pixel size of +\( 28\times 28 \), meaning we require \( 28 \times 28 = 724 \) weights to each +neuron in the first hidden layer. + +

    +If we were to analyze images of size \( 128\times 128 \) we would require +\( 128 \times 128 = 16384 \) weights to each neuron. Even worse if we were +dealing with color images, as most images are, we have an image matrix +of size \( 128\times 128 \) for each color dimension (Red, Green, Blue), +meaning 3 times the number of weights \( = 49152 \) are required for every +single neuron in the first hidden layer. + +

    +









    + +

    Strong correlations

    +Images typically have strong local correlations, meaning that a small +part of the image varies little from its neighboring regions. If for +example we have an image of a blue car, we can roughly assume that a +small blue part of the image is surrounded by other blue regions. + +

    +Therefore, instead of connecting every single pixel to a neuron in the +first hidden layer, as we have previously done with deep neural +networks, we can instead connect each neuron to a small part of the +image (in all 3 RGB depth dimensions). The size of each small area is +fixed, and known as a receptive. + +

    + + +

    Layers of a CNN

    +The layers of a convolutional neural network arrange neurons in 3D: width, height and depth. +The input image is typically a square matrix of depth 3. + +

    +A convolution is performed on the image which outputs +a 3D volume of neurons. The weights to the input are arranged in a number of 2D matrices, known as filters. + +

    +Each filter slides along the input image, taking the dot product +between each small part of the image and the filter, in all depth +dimensions. This is then passed through a non-linear function, +typically the Rectified Linear (ReLu) function, which serves as the +activation of the neurons in the first convolutional layer. This is +further passed through a pooling layer, which reduces the size of the +convolutional layer, e.g. by taking the maximum or average across some +small regions, and this serves as input to the next convolutional +layer. + +

    +









    + +

    Systematic reduction

    + +

    +By systematically reducing the size of the input volume, through +convolution and pooling, the network should create representations of +small parts of the input, and then from them assemble representations +of larger areas. The final pooling layer is flattened to serve as +input to a hidden layer, such that each neuron in the final pooling +layer is connected to every single neuron in the hidden layer. This +then serves as input to the output layer, e.g. a softmax output for +classification. + +

    +









    + +

    Prerequisites: Collect and pre-process data

    +

    + + +

    # import necessary packages
+import numpy as np
+import matplotlib.pyplot as plt
+from sklearn import datasets
+
+
+# ensure the same random numbers appear every time
+np.random.seed(0)
+
+# display images in notebook
+%matplotlib inline
+plt.rcParams['figure.figsize'] = (12,12)
+
+
+# download MNIST dataset
+digits = datasets.load_digits()
+
+# define inputs and labels
+inputs = digits.images
+labels = digits.target
+
+# RGB images have a depth of 3
+# our images are grayscale so they should have a depth of 1
+inputs = inputs[:,:,:,np.newaxis]
+
+print("inputs = (n_inputs, pixel_width, pixel_height, depth) = " + str(inputs.shape))
+print("labels = (n_inputs) = " + str(labels.shape))
+
+
+# choose some random images to display
+n_inputs = len(inputs)
+indices = np.arange(n_inputs)
+random_indices = np.random.choice(indices, size=5)
+
+for i, image in enumerate(digits.images[random_indices]):
+    plt.subplot(1, 5, i+1)
+    plt.axis('off')
+    plt.imshow(image, cmap=plt.cm.gray_r, interpolation='nearest')
+    plt.title("Label: %d" % digits.target[random_indices[i]])
+plt.show()
+
    +

    +









    + +

    Importing Keras and Tensorflow

    +

    + + +

    from keras.utils import to_categorical
+from sklearn.model_selection import train_test_split
+
+# representation of labels
+labels = to_categorical(labels)
+
+# split into train and test data
+# one-liner from scikit-learn library
+train_size = 0.8
+test_size = 1 - train_size
+X_train, X_test, Y_train, Y_test = train_test_split(inputs, labels, train_size=train_size,
+                                                    test_size=test_size)
+
    +

    +









    + +

    Using TensorFlow backend

    + +

    +We need to define model and architecture and choose cost function and optmizer. +

    + + +

    import tensorflow as tf
+
+class ConvolutionalNeuralNetworkTensorflow:
+    def __init__(
+            self,
+            X_train,
+            Y_train,
+            X_test,
+            Y_test,
+            n_filters=10,
+            n_neurons_connected=50,
+            n_categories=10,
+            receptive_field=3,
+            stride=1,
+            padding=1,
+            epochs=10,
+            batch_size=100,
+            eta=0.1,
+            lmbd=0.0):
+        
+        self.global_step = tf.Variable(0, dtype=tf.int32, trainable=False, name='global_step')
+        
+        self.X_train = X_train
+        self.Y_train = Y_train
+        self.X_test = X_test
+        self.Y_test = Y_test
+        
+        self.n_inputs, self.input_width, self.input_height, self.depth = X_train.shape
+        
+        self.n_filters = n_filters
+        self.n_downsampled = int(self.input_width*self.input_height*n_filters / 4)
+        self.n_neurons_connected = n_neurons_connected
+        self.n_categories = n_categories
+        
+        self.receptive_field = receptive_field
+        self.stride = stride
+        self.strides = [stride, stride, stride, stride]
+        self.padding = padding
+        
+        self.epochs = epochs
+        self.batch_size = batch_size
+        self.iterations = self.n_inputs // self.batch_size
+        self.eta = eta
+        self.lmbd = lmbd
+        
+        self.create_placeholders()
+        self.create_CNN()
+        self.create_loss()
+        self.create_optimiser()
+        self.create_accuracy()
+    
+    def create_placeholders(self):
+        with tf.name_scope('data'):
+            self.X = tf.placeholder(tf.float32, shape=(None, self.input_width, self.input_height, self.depth), name='X_data')
+            self.Y = tf.placeholder(tf.float32, shape=(None, self.n_categories), name='Y_data')
+    
+    def create_CNN(self):
+        with tf.name_scope('CNN'):
+            
+            # Convolutional layer
+            self.W_conv = self.weight_variable([self.receptive_field, self.receptive_field, self.depth, self.n_filters], name='conv', dtype=tf.float32)
+            b_conv = self.weight_variable([self.n_filters], name='conv', dtype=tf.float32)
+            z_conv = tf.nn.conv2d(self.X, self.W_conv, self.strides, padding='SAME', name='conv') + b_conv
+            a_conv = tf.nn.relu(z_conv)
+            
+            # 2x2 max pooling
+            a_pool = tf.nn.max_pool(a_conv, [1, 2, 2, 1], [1, 2, 2, 1], padding='SAME', name='pool')
+            
+            # Fully connected layer
+            a_pool_flat = tf.reshape(a_pool, [-1, self.n_downsampled])
+            self.W_fc = self.weight_variable([self.n_downsampled, self.n_neurons_connected], name='fc', dtype=tf.float32)
+            b_fc = self.bias_variable([self.n_neurons_connected], name='fc', dtype=tf.float32)
+            a_fc = tf.nn.relu(tf.matmul(a_pool_flat, self.W_fc) + b_fc)
+            
+            # Output layer
+            self.W_out = self.weight_variable([self.n_neurons_connected, self.n_categories], name='out', dtype=tf.float32)
+            b_out = self.bias_variable([self.n_categories], name='out', dtype=tf.float32)
+            self.z_out = tf.matmul(a_fc, self.W_out) + b_out
+    
+    def create_loss(self):
+        with tf.name_scope('loss'):
+            softmax_loss = tf.reduce_mean(tf.nn.softmax_cross_entropy_with_logits_v2(labels=self.Y, logits=self.z_out))
+            
+            regularizer_loss_conv = tf.nn.l2_loss(self.W_conv)
+            regularizer_loss_fc = tf.nn.l2_loss(self.W_fc)
+            regularizer_loss_out = tf.nn.l2_loss(self.W_out)
+            regularizer_loss = self.lmbd*(regularizer_loss_conv + regularizer_loss_fc + regularizer_loss_out)
+            
+            self.loss = softmax_loss + regularizer_loss
+
+    def create_accuracy(self):
+        with tf.name_scope('accuracy'):
+            probabilities = tf.nn.softmax(self.z_out)
+            predictions = tf.argmax(probabilities, 1)
+            labels = tf.argmax(self.Y, 1)
+            
+            correct_predictions = tf.equal(predictions, labels)
+            correct_predictions = tf.cast(correct_predictions, tf.float32)
+            self.accuracy = tf.reduce_mean(correct_predictions)
+    
+    def create_optimiser(self):
+        with tf.name_scope('optimizer'):
+            self.optimizer = tf.train.GradientDescentOptimizer(learning_rate=self.eta).minimize(self.loss, global_step=self.global_step)
+            
+    def weight_variable(self, shape, name='', dtype=tf.float32):
+        initial = tf.truncated_normal(shape, stddev=0.1)
+        return tf.Variable(initial, name=name, dtype=dtype)
+    
+    def bias_variable(self, shape, name='', dtype=tf.float32):
+        initial = tf.constant(0.1, shape=shape)
+        return tf.Variable(initial, name=name, dtype=dtype)
+
+    def fit(self):
+        data_indices = np.arange(self.n_inputs)
+
+        with tf.Session() as sess:
+            sess.run(tf.global_variables_initializer())
+            for i in range(self.epochs):
+                for j in range(self.iterations):
+                    chosen_datapoints = np.random.choice(data_indices, size=self.batch_size, replace=False)
+                    batch_X, batch_Y = self.X_train[chosen_datapoints], self.Y_train[chosen_datapoints]
+            
+                    sess.run([CNN.loss, CNN.optimizer],
+                        feed_dict={CNN.X: batch_X,
+                                   CNN.Y: batch_Y})
+                    accuracy = sess.run(CNN.accuracy,
+                        feed_dict={CNN.X: batch_X,
+                                   CNN.Y: batch_Y})
+                    step = sess.run(CNN.global_step)
+    
+            self.train_loss, self.train_accuracy = sess.run([CNN.loss, CNN.accuracy],
+                feed_dict={CNN.X: self.X_train,
+                           CNN.Y: self.Y_train})
+        
+            self.test_loss, self.test_accuracy = sess.run([CNN.loss, CNN.accuracy],
+                feed_dict={CNN.X: self.X_test,
+                           CNN.Y: self.Y_test})
+
    +

    +









    + +

    Train the model

    + +

    +We need now to train the model, evaluate it and test its performance on test data, and eventually include hyperparameters. +

    + + +

    epochs = 100
+batch_size = 100
+n_filters = 10
+n_neurons_connected = 50
+n_categories = 10
+
+eta_vals = np.logspace(-5, 1, 7)
+lmbd_vals = np.logspace(-5, 1, 7)
+CNN_tf = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)
+        
+for i, eta in enumerate(eta_vals):
+    for j, lmbd in enumerate(lmbd_vals):
+        CNN = ConvolutionalNeuralNetworkTensorflow(X_train, Y_train, X_test, Y_test,
+                                      n_filters=n_filters, n_neurons_connected=n_neurons_connected,
+                                      n_categories=n_categories, epochs=epochs, batch_size=batch_size,
+                                      eta=eta, lmbd=lmbd)
+        CNN.fit()
+        
+        print("Learning rate = ", eta)
+        print("Lambda = ", lmbd)
+        print("Test accuracy: %.3f" % CNN.test_accuracy)
+        print()
+            
+        CNN_tf[i][j] = CNN
+
    +

    +









    + +

    Visualizing the results

    + +

    + + +

    # visual representation of grid search
+# uses seaborn heatmap, could probably do this in matplotlib
+import seaborn as sns
+
+sns.set()
+
+train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
+test_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
+
+for i in range(len(eta_vals)):
+    for j in range(len(lmbd_vals)):
+        CNN = CNN_tf[i][j]
+
+        train_accuracy[i][j] = CNN.train_accuracy
+        test_accuracy[i][j] = CNN.test_accuracy
+
+        
+fig, ax = plt.subplots(figsize = (10, 10))
+sns.heatmap(train_accuracy, annot=True, ax=ax, cmap="viridis")
+ax.set_title("Training Accuracy")
+ax.set_ylabel("$\eta$")
+ax.set_xlabel("$\lambda$")
+plt.show()
+
+fig, ax = plt.subplots(figsize = (10, 10))
+sns.heatmap(test_accuracy, annot=True, ax=ax, cmap="viridis")
+ax.set_title("Test Accuracy")
+ax.set_ylabel("$\eta$")
+ax.set_xlabel("$\lambda$")
+plt.show()
+
    +

    + + +

    Running with Keras

    + +

    + + +

    from keras.models import Sequential
+from keras.layers.convolutional import Conv2D
+from keras.layers.convolutional import MaxPooling2D
+from keras.layers import Flatten
+from keras.layers import Dense
+from keras.regularizers import l2
+from keras.optimizers import SGD
+
+def create_convolutional_neural_network_keras(input_shape, receptive_field,
+                                              n_filters, n_neurons_connected, n_categories,
+                                              eta, lmbd):
+    model = Sequential()
+    model.add(Conv2D(n_filters, (receptive_field, receptive_field), input_shape=input_shape, padding='same',
+              activation='relu', kernel_regularizer=l2(lmbd)))
+    model.add(MaxPooling2D(pool_size=(2, 2)))
+    model.add(Flatten())
+    model.add(Dense(n_neurons_connected, activation='relu', kernel_regularizer=l2(lmbd)))
+    model.add(Dense(n_categories, activation='softmax', kernel_regularizer=l2(lmbd)))
+    
+    sgd = SGD(lr=eta)
+    model.compile(loss='categorical_crossentropy', optimizer=sgd, metrics=['accuracy'])
+    
+    return model
+
+epochs = 100
+batch_size = 100
+input_shape = X_train.shape[1:4]
+receptive_field = 3
+n_filters = 10
+n_neurons_connected = 50
+n_categories = 10
+
+eta_vals = np.logspace(-5, 1, 7)
+lmbd_vals = np.logspace(-5, 1, 7)
+
    +

    +









    + +

    Final part

    + +

    + + +

    CNN_keras = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)
+        
+for i, eta in enumerate(eta_vals):
+    for j, lmbd in enumerate(lmbd_vals):
+        CNN = create_convolutional_neural_network_keras(input_shape, receptive_field,
+                                              n_filters, n_neurons_connected, n_categories,
+                                              eta, lmbd)
+        CNN.fit(X_train, Y_train, epochs=epochs, batch_size=batch_size, verbose=0)
+        scores = CNN.evaluate(X_test, Y_test)
+        
+        CNN_keras[i][j] = CNN
+        
+        print("Learning rate = ", eta)
+        print("Lambda = ", lmbd)
+        print("Test accuracy: %.3f" % scores[1])
+        print()
+
    +

    +









    + +

    Final visualization

    + +

    + + +

    # visual representation of grid search
+# uses seaborn heatmap, could probably do this in matplotlib
+import seaborn as sns
+
+sns.set()
+
+train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
+test_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
+
+for i in range(len(eta_vals)):
+    for j in range(len(lmbd_vals)):
+        CNN = CNN_keras[i][j]
+
+        train_accuracy[i][j] = CNN.evaluate(X_train, Y_train)[1]
+        test_accuracy[i][j] = CNN.evaluate(X_test, Y_test)[1]
+
+        
+fig, ax = plt.subplots(figsize = (10, 10))
+sns.heatmap(train_accuracy, annot=True, ax=ax, cmap="viridis")
+ax.set_title("Training Accuracy")
+ax.set_ylabel("$\eta$")
+ax.set_xlabel("$\lambda$")
+plt.show()
+
+fig, ax = plt.subplots(figsize = (10, 10))
+sns.heatmap(test_accuracy, annot=True, ax=ax, cmap="viridis")
+ax.set_title("Test Accuracy")
+ax.set_ylabel("$\eta$")
+ax.set_xlabel("$\lambda$")
+plt.show()
+
    +

    +









    + +

    Fun links

    + +
      +
    1. Self-Driving cars using a convolutional neural network
      2. +
    2. Abstract art using convolutional neural networks
      3. +
    + + + + + +
    + © 1999-2019, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license +
    + + + + + + diff --git a/doc/src/CNN/cnn.ipynb b/doc/src/CNN/cnn.ipynb new file mode 100644 index 000000000..07b751bcb --- /dev/null +++ b/doc/src/CNN/cnn.ipynb @@ -0,0 +1,729 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "# Convolutional Neural Networks\n", + "\n", + " \n", + "**Morten Hjorth-Jensen**, Department of Physics, University of Oslo and Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University\n", + "\n", + "Date: **Oct 3, 2019**\n", + "\n", + "Copyright 1999-2019, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "## Convolutional Neural Networks (recognizing images)\n", + "\n", + "Convolutional Neural Networks (CNN) are very similar to ordinary Neural Networks.\n", + "\n", + "They are made up of neurons that have learnable weights and\n", + "biases. Each neuron receives some inputs, performs a dot product and\n", + "optionally follows it with a non-linearity. The whole network still\n", + "expresses a single differentiable score function: from the raw image\n", + "pixels on one end to class scores at the other. And they still have a\n", + "loss function (for example Softmax) on the last (fully-connected) layer\n", + "and all the tips/tricks we developed for learning regular Neural\n", + "Networks still apply (back propagation, gradient descent etc etc).\n", + "\n", + "What is the difference? **CNN architectures make the explicit assumption that\n", + "the inputs are images, which allows us to encode certain properties\n", + "into the architecture. These then make the forward function more\n", + "efficient to implement and vastly reduce the amount of parameters in\n", + "the network.**\n", + "\n", + "Here we provide only a superficial overview, for the more interested, we recommend highly the course\n", + "[IN5400 – Machine Learning for Image Analysis](https://www.uio.no/studier/emner/matnat/ifi/IN5400/index-eng.html)\n", + "and the slides of [CS231](http://cs231n.github.io/convolutional-networks/).\n", + "\n", + "## Regular NNs don’t scale well to full images\n", + "\n", + "As an example, consider\n", + "an image of size $32\\times 32\\times 3$ (32 wide, 32 high, 3 color channels), so a\n", + "single fully-connected neuron in a first hidden layer of a regular\n", + "Neural Network would have $32\\times 32\\times 3 = 3072$ weights. This amount still\n", + "seems manageable, but clearly this fully-connected structure does not\n", + "scale to larger images. For example, an image of more respectable\n", + "size, say $200\\times 200\\times 3$, would lead to neurons that have \n", + "$200\\times 200\\times 3 = 120,000$ weights. \n", + "\n", + "We could have\n", + "several such neurons, and the parameters would add up quickly! Clearly,\n", + "this full connectivity is wasteful and the huge number of parameters\n", + "would quickly lead to possible overfitting.\n", + "\n", + "\n", + "\n", + "\n", + "

    A regular 3-layer Neural Network.

    \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "## 3D volumes of neurons\n", + "\n", + "Convolutional Neural Networks take advantage of the fact that the\n", + "input consists of images and they constrain the architecture in a more\n", + "sensible way. \n", + "\n", + "In particular, unlike a regular Neural Network, the\n", + "layers of a CNN have neurons arranged in 3 dimensions: width,\n", + "height, depth. (Note that the word depth here refers to the third\n", + "dimension of an activation volume, not to the depth of a full Neural\n", + "Network, which can refer to the total number of layers in a network.)\n", + "\n", + "To understand it better, the above example of an image \n", + "with an input volume of\n", + "activations has dimensions $32\\times 32\\times 3$ (width, height,\n", + "depth respectively). \n", + "\n", + "The neurons in a layer will\n", + "only be connected to a small region of the layer before it, instead of\n", + "all of the neurons in a fully-connected manner. Moreover, the final\n", + "output layer could for this specific image have dimensions $1\\times 1 \\times 10$, \n", + "because by the\n", + "end of the CNN architecture we will reduce the full image into a\n", + "single vector of class scores, arranged along the depth\n", + "dimension. \n", + "\n", + "\n", + "\n", + "\n", + "

    A CNN arranges its neurons in three dimensions (width, height, depth), as visualized in one of the layers. Every layer of a CNN transforms the 3D input volume to a 3D output volume of neuron activations. In this example, the red input layer holds the image, so its width and height would be the dimensions of the image, and the depth would be 3 (Red, Green, Blue channels).

    \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "## Layers used to build CNNs\n", + "\n", + "\n", + "A simple CNN is a sequence of layers, and every layer of a CNN\n", + "transforms one volume of activations to another through a\n", + "differentiable function. We use three main types of layers to build\n", + "CNN architectures: Convolutional Layer, Pooling Layer, and\n", + "Fully-Connected Layer (exactly as seen in regular Neural Networks). We\n", + "will stack these layers to form a full CNN architecture.\n", + "\n", + "A simple CNN for image classification could have the architecture:\n", + "\n", + "* **INPUT** ($32\\times 32 \\times 3$) will hold the raw pixel values of the image, in this case an image of width 32, height 32, and with three color channels R,G,B.\n", + "\n", + "* **CONV** (convolutional )layer will compute the output of neurons that are connected to local regions in the input, each computing a dot product between their weights and a small region they are connected to in the input volume. This may result in volume such as $[32\\times 32\\times 12]$ if we decided to use 12 filters.\n", + "\n", + "* **RELU** layer will apply an elementwise activation function, such as the $max(0,x)$ thresholding at zero. This leaves the size of the volume unchanged ($[32\\times 32\\times 12]$).\n", + "\n", + "* **POOL** (pooling) layer will perform a downsampling operation along the spatial dimensions (width, height), resulting in volume such as $[16\\times 16\\times 12]$.\n", + "\n", + "* **FC** (i.e. fully-connected) layer will compute the class scores, resulting in volume of size $[1\\times 1\\times 10]$, where each of the 10 numbers correspond to a class score, such as among the 10 categories of the MNIST images we considered above . As with ordinary Neural Networks and as the name implies, each neuron in this layer will be connected to all the numbers in the previous volume.\n", + "\n", + "## Transforming images\n", + "\n", + "CNNs transform the original image layer by layer from the original\n", + "pixel values to the final class scores. \n", + "\n", + "Observe that some layers contain\n", + "parameters and other don’t. In particular, the CNN layers perform\n", + "transformations that are a function of not only the activations in the\n", + "input volume, but also of the parameters (the weights and biases of\n", + "the neurons). On the other hand, the RELU/POOL layers will implement a\n", + "fixed function. The parameters in the CONV/FC layers will be trained\n", + "with gradient descent so that the class scores that the CNN computes\n", + "are consistent with the labels in the training set for each image.\n", + "\n", + "\n", + "## CNNs in brief\n", + "\n", + "In summary:\n", + "\n", + "* A CNN architecture is in the simplest case a list of Layers that transform the image volume into an output volume (e.g. holding the class scores)\n", + "\n", + "* There are a few distinct types of Layers (e.g. CONV/FC/RELU/POOL are by far the most popular)\n", + "\n", + "* Each Layer accepts an input 3D volume and transforms it to an output 3D volume through a differentiable function\n", + "\n", + "* Each Layer may or may not have parameters (e.g. CONV/FC do, RELU/POOL don’t)\n", + "\n", + "* Each Layer may or may not have additional hyperparameters (e.g. CONV/FC/POOL do, RELU doesn’t)\n", + "\n", + "For more material on convolutional networks, we strongly recommend\n", + "the course\n", + "[IN5400 – Machine Learning for Image Analysis](https://www.uio.no/studier/emner/matnat/ifi/IN5400/index-eng.html)\n", + "and the slides of [CS231](http://cs231n.github.io/convolutional-networks/) which is taught at Stanford University (consistently ranked as one of the top computer science programs in the world). [Michael Nielsen's book is a must read, in particular chapter 6 which deals with CNNs](http://neuralnetworksanddeeplearning.com/chap6.html).\n", + "\n", + "\n", + "## CNNs in more detail, building convolutional neural networks in Tensorflow and Keras\n", + "\n", + "\n", + "As discussed above, CNNs are neural networks built from the assumption that the inputs\n", + "to the network are 2D images. This is important because the number of features or pixels in images\n", + "grows very fast with the image size, and an enormous number of weights and biases are needed in order to build an accurate network. \n", + "\n", + "As before, we still have our input, a hidden layer and an output. What's novel about convolutional networks\n", + "are the **convolutional** and **pooling** layers stacked in pairs between the input and the hidden layer.\n", + "In addition, the data is no longer represented as a 2D feature matrix, instead each input is a number of 2D\n", + "matrices, typically 1 for each color dimension (Red, Green, Blue). \n", + "\n", + "\n", + "## Setting it up\n", + "\n", + "It means that to represent the entire\n", + "dataset of images, we require a 4D matrix or **tensor**. This tensor has the dimensions:" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "(n_{inputs},\\, n_{pixels, width},\\, n_{pixels, height},\\, depth) .\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## The MNIST dataset again\n", + "\n", + "The MNIST dataset consists of grayscale images with a pixel size of\n", + "$28\\times 28$, meaning we require $28 \\times 28 = 724$ weights to each\n", + "neuron in the first hidden layer.\n", + "\n", + "If we were to analyze images of size $128\\times 128$ we would require\n", + "$128 \\times 128 = 16384$ weights to each neuron. Even worse if we were\n", + "dealing with color images, as most images are, we have an image matrix\n", + "of size $128\\times 128$ for each color dimension (Red, Green, Blue),\n", + "meaning 3 times the number of weights $= 49152$ are required for every\n", + "single neuron in the first hidden layer.\n", + "\n", + "\n", + "## Strong correlations\n", + "Images typically have strong local correlations, meaning that a small\n", + "part of the image varies little from its neighboring regions. If for\n", + "example we have an image of a blue car, we can roughly assume that a\n", + "small blue part of the image is surrounded by other blue regions.\n", + "\n", + "Therefore, instead of connecting every single pixel to a neuron in the\n", + "first hidden layer, as we have previously done with deep neural\n", + "networks, we can instead connect each neuron to a small part of the\n", + "image (in all 3 RGB depth dimensions). The size of each small area is\n", + "fixed, and known as a [receptive](https://en.wikipedia.org/wiki/Receptive_field).\n", + "\n", + "\n", + "\n", + "## Layers of a CNN\n", + "The layers of a convolutional neural network arrange neurons in 3D: width, height and depth. \n", + "The input image is typically a square matrix of depth 3. \n", + "\n", + "A **convolution** is performed on the image which outputs\n", + "a 3D volume of neurons. The weights to the input are arranged in a number of 2D matrices, known as **filters**.\n", + "\n", + "\n", + "Each filter slides along the input image, taking the dot product\n", + "between each small part of the image and the filter, in all depth\n", + "dimensions. This is then passed through a non-linear function,\n", + "typically the **Rectified Linear (ReLu)** function, which serves as the\n", + "activation of the neurons in the first convolutional layer. This is\n", + "further passed through a **pooling layer**, which reduces the size of the\n", + "convolutional layer, e.g. by taking the maximum or average across some\n", + "small regions, and this serves as input to the next convolutional\n", + "layer.\n", + "\n", + "\n", + "## Systematic reduction\n", + "\n", + "By systematically reducing the size of the input volume, through\n", + "convolution and pooling, the network should create representations of\n", + "small parts of the input, and then from them assemble representations\n", + "of larger areas. The final pooling layer is flattened to serve as\n", + "input to a hidden layer, such that each neuron in the final pooling\n", + "layer is connected to every single neuron in the hidden layer. This\n", + "then serves as input to the output layer, e.g. a softmax output for\n", + "classification.\n", + "\n", + "\n", + "## Prerequisites: Collect and pre-process data" + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "%matplotlib inline\n", + "\n", + "# import necessary packages\n", + "import numpy as np\n", + "import matplotlib.pyplot as plt\n", + "from sklearn import datasets\n", + "\n", + "\n", + "# ensure the same random numbers appear every time\n", + "np.random.seed(0)\n", + "\n", + "# display images in notebook\n", + "%matplotlib inline\n", + "plt.rcParams['figure.figsize'] = (12,12)\n", + "\n", + "\n", + "# download MNIST dataset\n", + "digits = datasets.load_digits()\n", + "\n", + "# define inputs and labels\n", + "inputs = digits.images\n", + "labels = digits.target\n", + "\n", + "# RGB images have a depth of 3\n", + "# our images are grayscale so they should have a depth of 1\n", + "inputs = inputs[:,:,:,np.newaxis]\n", + "\n", + "print(\"inputs = (n_inputs, pixel_width, pixel_height, depth) = \" + str(inputs.shape))\n", + "print(\"labels = (n_inputs) = \" + str(labels.shape))\n", + "\n", + "\n", + "# choose some random images to display\n", + "n_inputs = len(inputs)\n", + "indices = np.arange(n_inputs)\n", + "random_indices = np.random.choice(indices, size=5)\n", + "\n", + "for i, image in enumerate(digits.images[random_indices]):\n", + " plt.subplot(1, 5, i+1)\n", + " plt.axis('off')\n", + " plt.imshow(image, cmap=plt.cm.gray_r, interpolation='nearest')\n", + " plt.title(\"Label: %d\" % digits.target[random_indices[i]])\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Importing Keras and Tensorflow" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "from keras.utils import to_categorical\n", + "from sklearn.model_selection import train_test_split\n", + "\n", + "# representation of labels\n", + "labels = to_categorical(labels)\n", + "\n", + "# split into train and test data\n", + "# one-liner from scikit-learn library\n", + "train_size = 0.8\n", + "test_size = 1 - train_size\n", + "X_train, X_test, Y_train, Y_test = train_test_split(inputs, labels, train_size=train_size,\n", + " test_size=test_size)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Using TensorFlow backend\n", + "\n", + "We need to define model and architecture and choose cost function and optmizer." + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "\n", + "import tensorflow as tf\n", + "\n", + "class ConvolutionalNeuralNetworkTensorflow:\n", + " def __init__(\n", + " self,\n", + " X_train,\n", + " Y_train,\n", + " X_test,\n", + " Y_test,\n", + " n_filters=10,\n", + " n_neurons_connected=50,\n", + " n_categories=10,\n", + " receptive_field=3,\n", + " stride=1,\n", + " padding=1,\n", + " epochs=10,\n", + " batch_size=100,\n", + " eta=0.1,\n", + " lmbd=0.0):\n", + " \n", + " self.global_step = tf.Variable(0, dtype=tf.int32, trainable=False, name='global_step')\n", + " \n", + " self.X_train = X_train\n", + " self.Y_train = Y_train\n", + " self.X_test = X_test\n", + " self.Y_test = Y_test\n", + " \n", + " self.n_inputs, self.input_width, self.input_height, self.depth = X_train.shape\n", + " \n", + " self.n_filters = n_filters\n", + " self.n_downsampled = int(self.input_width*self.input_height*n_filters / 4)\n", + " self.n_neurons_connected = n_neurons_connected\n", + " self.n_categories = n_categories\n", + " \n", + " self.receptive_field = receptive_field\n", + " self.stride = stride\n", + " self.strides = [stride, stride, stride, stride]\n", + " self.padding = padding\n", + " \n", + " self.epochs = epochs\n", + " self.batch_size = batch_size\n", + " self.iterations = self.n_inputs // self.batch_size\n", + " self.eta = eta\n", + " self.lmbd = lmbd\n", + " \n", + " self.create_placeholders()\n", + " self.create_CNN()\n", + " self.create_loss()\n", + " self.create_optimiser()\n", + " self.create_accuracy()\n", + " \n", + " def create_placeholders(self):\n", + " with tf.name_scope('data'):\n", + " self.X = tf.placeholder(tf.float32, shape=(None, self.input_width, self.input_height, self.depth), name='X_data')\n", + " self.Y = tf.placeholder(tf.float32, shape=(None, self.n_categories), name='Y_data')\n", + " \n", + " def create_CNN(self):\n", + " with tf.name_scope('CNN'):\n", + " \n", + " # Convolutional layer\n", + " self.W_conv = self.weight_variable([self.receptive_field, self.receptive_field, self.depth, self.n_filters], name='conv', dtype=tf.float32)\n", + " b_conv = self.weight_variable([self.n_filters], name='conv', dtype=tf.float32)\n", + " z_conv = tf.nn.conv2d(self.X, self.W_conv, self.strides, padding='SAME', name='conv') + b_conv\n", + " a_conv = tf.nn.relu(z_conv)\n", + " \n", + " # 2x2 max pooling\n", + " a_pool = tf.nn.max_pool(a_conv, [1, 2, 2, 1], [1, 2, 2, 1], padding='SAME', name='pool')\n", + " \n", + " # Fully connected layer\n", + " a_pool_flat = tf.reshape(a_pool, [-1, self.n_downsampled])\n", + " self.W_fc = self.weight_variable([self.n_downsampled, self.n_neurons_connected], name='fc', dtype=tf.float32)\n", + " b_fc = self.bias_variable([self.n_neurons_connected], name='fc', dtype=tf.float32)\n", + " a_fc = tf.nn.relu(tf.matmul(a_pool_flat, self.W_fc) + b_fc)\n", + " \n", + " # Output layer\n", + " self.W_out = self.weight_variable([self.n_neurons_connected, self.n_categories], name='out', dtype=tf.float32)\n", + " b_out = self.bias_variable([self.n_categories], name='out', dtype=tf.float32)\n", + " self.z_out = tf.matmul(a_fc, self.W_out) + b_out\n", + " \n", + " def create_loss(self):\n", + " with tf.name_scope('loss'):\n", + " softmax_loss = tf.reduce_mean(tf.nn.softmax_cross_entropy_with_logits_v2(labels=self.Y, logits=self.z_out))\n", + " \n", + " regularizer_loss_conv = tf.nn.l2_loss(self.W_conv)\n", + " regularizer_loss_fc = tf.nn.l2_loss(self.W_fc)\n", + " regularizer_loss_out = tf.nn.l2_loss(self.W_out)\n", + " regularizer_loss = self.lmbd*(regularizer_loss_conv + regularizer_loss_fc + regularizer_loss_out)\n", + " \n", + " self.loss = softmax_loss + regularizer_loss\n", + "\n", + " def create_accuracy(self):\n", + " with tf.name_scope('accuracy'):\n", + " probabilities = tf.nn.softmax(self.z_out)\n", + " predictions = tf.argmax(probabilities, 1)\n", + " labels = tf.argmax(self.Y, 1)\n", + " \n", + " correct_predictions = tf.equal(predictions, labels)\n", + " correct_predictions = tf.cast(correct_predictions, tf.float32)\n", + " self.accuracy = tf.reduce_mean(correct_predictions)\n", + " \n", + " def create_optimiser(self):\n", + " with tf.name_scope('optimizer'):\n", + " self.optimizer = tf.train.GradientDescentOptimizer(learning_rate=self.eta).minimize(self.loss, global_step=self.global_step)\n", + " \n", + " def weight_variable(self, shape, name='', dtype=tf.float32):\n", + " initial = tf.truncated_normal(shape, stddev=0.1)\n", + " return tf.Variable(initial, name=name, dtype=dtype)\n", + " \n", + " def bias_variable(self, shape, name='', dtype=tf.float32):\n", + " initial = tf.constant(0.1, shape=shape)\n", + " return tf.Variable(initial, name=name, dtype=dtype)\n", + "\n", + " def fit(self):\n", + " data_indices = np.arange(self.n_inputs)\n", + "\n", + " with tf.Session() as sess:\n", + " sess.run(tf.global_variables_initializer())\n", + " for i in range(self.epochs):\n", + " for j in range(self.iterations):\n", + " chosen_datapoints = np.random.choice(data_indices, size=self.batch_size, replace=False)\n", + " batch_X, batch_Y = self.X_train[chosen_datapoints], self.Y_train[chosen_datapoints]\n", + " \n", + " sess.run([CNN.loss, CNN.optimizer],\n", + " feed_dict={CNN.X: batch_X,\n", + " CNN.Y: batch_Y})\n", + " accuracy = sess.run(CNN.accuracy,\n", + " feed_dict={CNN.X: batch_X,\n", + " CNN.Y: batch_Y})\n", + " step = sess.run(CNN.global_step)\n", + " \n", + " self.train_loss, self.train_accuracy = sess.run([CNN.loss, CNN.accuracy],\n", + " feed_dict={CNN.X: self.X_train,\n", + " CNN.Y: self.Y_train})\n", + " \n", + " self.test_loss, self.test_accuracy = sess.run([CNN.loss, CNN.accuracy],\n", + " feed_dict={CNN.X: self.X_test,\n", + " CNN.Y: self.Y_test})" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Train the model\n", + "\n", + "We need now to train the model, evaluate it and test its performance on test data, and eventually include hyperparameters." + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "epochs = 100\n", + "batch_size = 100\n", + "n_filters = 10\n", + "n_neurons_connected = 50\n", + "n_categories = 10\n", + "\n", + "eta_vals = np.logspace(-5, 1, 7)\n", + "lmbd_vals = np.logspace(-5, 1, 7)\n", + "CNN_tf = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)\n", + " \n", + "for i, eta in enumerate(eta_vals):\n", + " for j, lmbd in enumerate(lmbd_vals):\n", + " CNN = ConvolutionalNeuralNetworkTensorflow(X_train, Y_train, X_test, Y_test,\n", + " n_filters=n_filters, n_neurons_connected=n_neurons_connected,\n", + " n_categories=n_categories, epochs=epochs, batch_size=batch_size,\n", + " eta=eta, lmbd=lmbd)\n", + " CNN.fit()\n", + " \n", + " print(\"Learning rate = \", eta)\n", + " print(\"Lambda = \", lmbd)\n", + " print(\"Test accuracy: %.3f\" % CNN.test_accuracy)\n", + " print()\n", + " \n", + " CNN_tf[i][j] = CNN" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Visualizing the results" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "# visual representation of grid search\n", + "# uses seaborn heatmap, could probably do this in matplotlib\n", + "import seaborn as sns\n", + "\n", + "sns.set()\n", + "\n", + "train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))\n", + "test_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))\n", + "\n", + "for i in range(len(eta_vals)):\n", + " for j in range(len(lmbd_vals)):\n", + " CNN = CNN_tf[i][j]\n", + "\n", + " train_accuracy[i][j] = CNN.train_accuracy\n", + " test_accuracy[i][j] = CNN.test_accuracy\n", + "\n", + " \n", + "fig, ax = plt.subplots(figsize = (10, 10))\n", + "sns.heatmap(train_accuracy, annot=True, ax=ax, cmap=\"viridis\")\n", + "ax.set_title(\"Training Accuracy\")\n", + "ax.set_ylabel(\"$\\eta$\")\n", + "ax.set_xlabel(\"$\\lambda$\")\n", + "plt.show()\n", + "\n", + "fig, ax = plt.subplots(figsize = (10, 10))\n", + "sns.heatmap(test_accuracy, annot=True, ax=ax, cmap=\"viridis\")\n", + "ax.set_title(\"Test Accuracy\")\n", + "ax.set_ylabel(\"$\\eta$\")\n", + "ax.set_xlabel(\"$\\lambda$\")\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "## Running with Keras" + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "from keras.models import Sequential\n", + "from keras.layers.convolutional import Conv2D\n", + "from keras.layers.convolutional import MaxPooling2D\n", + "from keras.layers import Flatten\n", + "from keras.layers import Dense\n", + "from keras.regularizers import l2\n", + "from keras.optimizers import SGD\n", + "\n", + "def create_convolutional_neural_network_keras(input_shape, receptive_field,\n", + " n_filters, n_neurons_connected, n_categories,\n", + " eta, lmbd):\n", + " model = Sequential()\n", + " model.add(Conv2D(n_filters, (receptive_field, receptive_field), input_shape=input_shape, padding='same',\n", + " activation='relu', kernel_regularizer=l2(lmbd)))\n", + " model.add(MaxPooling2D(pool_size=(2, 2)))\n", + " model.add(Flatten())\n", + " model.add(Dense(n_neurons_connected, activation='relu', kernel_regularizer=l2(lmbd)))\n", + " model.add(Dense(n_categories, activation='softmax', kernel_regularizer=l2(lmbd)))\n", + " \n", + " sgd = SGD(lr=eta)\n", + " model.compile(loss='categorical_crossentropy', optimizer=sgd, metrics=['accuracy'])\n", + " \n", + " return model\n", + "\n", + "epochs = 100\n", + "batch_size = 100\n", + "input_shape = X_train.shape[1:4]\n", + "receptive_field = 3\n", + "n_filters = 10\n", + "n_neurons_connected = 50\n", + "n_categories = 10\n", + "\n", + "eta_vals = np.logspace(-5, 1, 7)\n", + "lmbd_vals = np.logspace(-5, 1, 7)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Final part" + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "CNN_keras = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)\n", + " \n", + "for i, eta in enumerate(eta_vals):\n", + " for j, lmbd in enumerate(lmbd_vals):\n", + " CNN = create_convolutional_neural_network_keras(input_shape, receptive_field,\n", + " n_filters, n_neurons_connected, n_categories,\n", + " eta, lmbd)\n", + " CNN.fit(X_train, Y_train, epochs=epochs, batch_size=batch_size, verbose=0)\n", + " scores = CNN.evaluate(X_test, Y_test)\n", + " \n", + " CNN_keras[i][j] = CNN\n", + " \n", + " print(\"Learning rate = \", eta)\n", + " print(\"Lambda = \", lmbd)\n", + " print(\"Test accuracy: %.3f\" % scores[1])\n", + " print()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Final visualization" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + " # visual representation of grid search\n", + " # uses seaborn heatmap, could probably do this in matplotlib\n", + " import seaborn as sns\n", + " \n", + " sns.set()\n", + " \n", + " train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))\n", + " test_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))\n", + " \n", + " for i in range(len(eta_vals)):\n", + " for j in range(len(lmbd_vals)):\n", + " CNN = CNN_keras[i][j]\n", + " \n", + " train_accuracy[i][j] = CNN.evaluate(X_train, Y_train)[1]\n", + " test_accuracy[i][j] = CNN.evaluate(X_test, Y_test)[1]\n", + " \n", + " \n", + " fig, ax = plt.subplots(figsize = (10, 10))\n", + " sns.heatmap(train_accuracy, annot=True, ax=ax, cmap=\"viridis\")\n", + " ax.set_title(\"Training Accuracy\")\n", + " ax.set_ylabel(\"$\\eta$\")\n", + " ax.set_xlabel(\"$\\lambda$\")\n", + " plt.show()\n", + " \n", + " fig, ax = plt.subplots(figsize = (10, 10))\n", + " sns.heatmap(test_accuracy, annot=True, ax=ax, cmap=\"viridis\")\n", + " ax.set_title(\"Test Accuracy\")\n", + " ax.set_ylabel(\"$\\eta$\")\n", + " ax.set_xlabel(\"$\\lambda$\")\n", + " plt.show()\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Fun links\n", + "\n", + "1. [Self-Driving cars using a convolutional neural network](https://arxiv.org/abs/1604.07316)\n", + "\n", + "2. [Abstract art using convolutional neural networks](https://deepdreamgenerator.com/)" + ] + } + ], + "metadata": {}, + "nbformat": 4, + "nbformat_minor": 2 +} diff --git a/doc/src/CNN/cnn.p.tex b/doc/src/CNN/cnn.p.tex new file mode 100644 index 000000000..97e61e91d --- /dev/null +++ b/doc/src/CNN/cnn.p.tex @@ -0,0 +1,811 @@ +%% +%% Automatically generated file from DocOnce source +%% (https://github.com/hplgit/doconce/) +%% +%% +% #ifdef PTEX2TEX_EXPLANATION +%% +%% The file follows the ptex2tex extended LaTeX format, see +%% ptex2tex: http://code.google.com/p/ptex2tex/ +%% +%% Run +%% ptex2tex myfile +%% or +%% doconce ptex2tex myfile +%% +%% to turn myfile.p.tex into an ordinary LaTeX file myfile.tex. +%% (The ptex2tex program: http://code.google.com/p/ptex2tex) +%% Many preprocess options can be added to ptex2tex or doconce ptex2tex +%% +%% ptex2tex -DMINTED myfile +%% doconce ptex2tex myfile envir=minted +%% +%% ptex2tex will typeset code environments according to a global or local +%% .ptex2tex.cfg configure file. doconce ptex2tex will typeset code +%% according to options on the command line (just type doconce ptex2tex to +%% see examples). If doconce ptex2tex has envir=minted, it enables the +%% minted style without needing -DMINTED. +% #endif + +% #define PREAMBLE + +% #ifdef PREAMBLE +%-------------------- begin preamble ---------------------- + +\documentclass[% +oneside, % oneside: electronic viewing, twoside: printing +final, % draft: marks overfull hboxes, figures with paths +10pt]{article} + +\listfiles % print all files needed to compile this document + +\usepackage{relsize,makeidx,color,setspace,amsmath,amsfonts,amssymb} +\usepackage[table]{xcolor} +\usepackage{bm,ltablex,microtype} + +\usepackage[pdftex]{graphicx} + +\usepackage{ptex2tex} +% #ifdef MINTED +\usepackage{minted} +\usemintedstyle{default} +% #endif + +\usepackage[T1]{fontenc} +%\usepackage[latin1]{inputenc} +\usepackage{ucs} +\usepackage[utf8x]{inputenc} + +\usepackage{lmodern} % Latin Modern fonts derived from Computer Modern + +% Hyperlinks in PDF: +\definecolor{linkcolor}{rgb}{0,0,0.4} +\usepackage{hyperref} +\hypersetup{ + breaklinks=true, + colorlinks=true, + linkcolor=linkcolor, + urlcolor=linkcolor, + citecolor=black, + filecolor=black, + %filecolor=blue, + pdfmenubar=true, + pdftoolbar=true, + bookmarksdepth=3 % Uncomment (and tweak) for PDF bookmarks with more levels than the TOC + } +%\hyperbaseurl{} % hyperlinks are relative to this root + +\setcounter{tocdepth}{2} % levels in table of contents + +% Tricks for having figures close to where they are defined: +% 1. define less restrictive rules for where to put figures +\setcounter{topnumber}{2} +\setcounter{bottomnumber}{2} +\setcounter{totalnumber}{4} +\renewcommand{\topfraction}{0.95} +\renewcommand{\bottomfraction}{0.95} +\renewcommand{\textfraction}{0} +\renewcommand{\floatpagefraction}{0.75} +% floatpagefraction must always be less than topfraction! +% 2. ensure all figures are flushed before next section +\usepackage[section]{placeins} +% 3. enable begin{figure}[H] (often leads to ugly pagebreaks) +%\usepackage{float}\restylefloat{figure} + +% --- fancyhdr package for fancy headers --- +\usepackage{fancyhdr} +\fancyhf{} % sets both header and footer to nothing +\renewcommand{\headrulewidth}{0pt} +\fancyfoot[LE,RO]{\thepage} +% Ensure copyright on titlepage (article style) and chapter pages (book style) +\fancypagestyle{plain}{ + \fancyhf{} + \fancyfoot[C]{{\footnotesize \copyright\ 1999-2019, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license}} +% \renewcommand{\footrulewidth}{0mm} + \renewcommand{\headrulewidth}{0mm} +} +% Ensure copyright on titlepages with \thispagestyle{empty} +\fancypagestyle{empty}{ + \fancyhf{} + \fancyfoot[C]{{\footnotesize \copyright\ 1999-2019, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license}} + \renewcommand{\footrulewidth}{0mm} + \renewcommand{\headrulewidth}{0mm} +} + +\pagestyle{fancy} + + +% prevent orhpans and widows +\clubpenalty = 10000 +\widowpenalty = 10000 + +% --- end of standard preamble for documents --- + + +% insert custom LaTeX commands... + +\raggedbottom +\makeindex +\usepackage[totoc]{idxlayout} % for index in the toc +\usepackage[nottoc]{tocbibind} % for references/bibliography in the toc + +%-------------------- end preamble ---------------------- + +\begin{document} + +% matching end for #ifdef PREAMBLE +% #endif + +\newcommand{\exercisesection}[1]{\subsection*{#1}} + + +% ------------------- main content ---------------------- + + + +% ----------------- title ------------------------- + +\thispagestyle{empty} + +\begin{center} +{\LARGE\bf +\begin{spacing}{1.25} +Convolutional Neural Networks +\end{spacing} +} +\end{center} + +% ----------------- author(s) ------------------------- + +\begin{center} +{\bf Morten Hjorth-Jensen${}^{1, 2}$} \\ [0mm] +\end{center} + +\begin{center} +% List of all institutions: +\centerline{{\small ${}^1$Department of Physics, University of Oslo}} +\centerline{{\small ${}^2$Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University}} +\end{center} + +% ----------------- end author(s) ------------------------- + +% --- begin date --- +\begin{center} +Oct 3, 2019 +\end{center} +% --- end date --- + +\vspace{1cm} + + +% !split +\subsection{Convolutional Neural Networks (recognizing images)} + +Convolutional Neural Networks (CNN) are very similar to ordinary Neural Networks. + +They are made up of neurons that have learnable weights and +biases. Each neuron receives some inputs, performs a dot product and +optionally follows it with a non-linearity. The whole network still +expresses a single differentiable score function: from the raw image +pixels on one end to class scores at the other. And they still have a +loss function (for example Softmax) on the last (fully-connected) layer +and all the tips/tricks we developed for learning regular Neural +Networks still apply (back propagation, gradient descent etc etc). + +What is the difference? \textbf{CNN architectures make the explicit assumption that +the inputs are images, which allows us to encode certain properties +into the architecture. These then make the forward function more +efficient to implement and vastly reduce the amount of parameters in +the network.} + +Here we provide only a superficial overview, for the more interested, we recommend highly the course +\href{{https://www.uio.no/studier/emner/matnat/ifi/IN5400/index-eng.html}}{IN5400 – Machine Learning for Image Analysis} +and the slides of \href{{http://cs231n.github.io/convolutional-networks/}}{CS231}. + +% !split +\subsection{Regular NNs don’t scale well to full images} + +As an example, consider +an image of size $32\times 32\times 3$ (32 wide, 32 high, 3 color channels), so a +single fully-connected neuron in a first hidden layer of a regular +Neural Network would have $32\times 32\times 3 = 3072$ weights. This amount still +seems manageable, but clearly this fully-connected structure does not +scale to larger images. For example, an image of more respectable +size, say $200\times 200\times 3$, would lead to neurons that have +$200\times 200\times 3 = 120,000$ weights. + +We could have +several such neurons, and the parameters would add up quickly! Clearly, +this full connectivity is wasteful and the huge number of parameters +would quickly lead to possible overfitting. + + +\begin{figure}[!ht] % + \centerline{\includegraphics[width=0.6\linewidth]{figslides/nn.jpeg}} + \caption{ + A regular 3-layer Neural Network. + } +\end{figure} +%\clearpage % flush figures + + +% !split +\subsection{3D volumes of neurons} + +Convolutional Neural Networks take advantage of the fact that the +input consists of images and they constrain the architecture in a more +sensible way. + +In particular, unlike a regular Neural Network, the +layers of a CNN have neurons arranged in 3 dimensions: width, +height, depth. (Note that the word depth here refers to the third +dimension of an activation volume, not to the depth of a full Neural +Network, which can refer to the total number of layers in a network.) + +To understand it better, the above example of an image +with an input volume of +activations has dimensions $32\times 32\times 3$ (width, height, +depth respectively). + +The neurons in a layer will +only be connected to a small region of the layer before it, instead of +all of the neurons in a fully-connected manner. Moreover, the final +output layer could for this specific image have dimensions $1\times 1 \times 10$, +because by the +end of the CNN architecture we will reduce the full image into a +single vector of class scores, arranged along the depth +dimension. + + +\begin{figure}[!ht] % + \centerline{\includegraphics[width=0.6\linewidth]{figslides/cnn.jpeg}} + \caption{ + A CNN arranges its neurons in three dimensions (width, height, depth), as visualized in one of the layers. Every layer of a CNN transforms the 3D input volume to a 3D output volume of neuron activations. In this example, the red input layer holds the image, so its width and height would be the dimensions of the image, and the depth would be 3 (Red, Green, Blue channels). + } +\end{figure} +%\clearpage % flush figures + + + + +% !split +\subsection{Layers used to build CNNs} + + +A simple CNN is a sequence of layers, and every layer of a CNN +transforms one volume of activations to another through a +differentiable function. We use three main types of layers to build +CNN architectures: Convolutional Layer, Pooling Layer, and +Fully-Connected Layer (exactly as seen in regular Neural Networks). We +will stack these layers to form a full CNN architecture. + +A simple CNN for image classification could have the architecture: + +\begin{itemize} +\item \textbf{INPUT} ($32\times 32 \times 3$) will hold the raw pixel values of the image, in this case an image of width 32, height 32, and with three color channels R,G,B. + +\item \textbf{CONV} (convolutional )layer will compute the output of neurons that are connected to local regions in the input, each computing a dot product between their weights and a small region they are connected to in the input volume. This may result in volume such as $[32\times 32\times 12]$ if we decided to use 12 filters. + +\item \textbf{RELU} layer will apply an elementwise activation function, such as the $max(0,x)$ thresholding at zero. This leaves the size of the volume unchanged ($[32\times 32\times 12]$). + +\item \textbf{POOL} (pooling) layer will perform a downsampling operation along the spatial dimensions (width, height), resulting in volume such as $[16\times 16\times 12]$. + +\item \textbf{FC} (i.e.~fully-connected) layer will compute the class scores, resulting in volume of size $[1\times 1\times 10]$, where each of the 10 numbers correspond to a class score, such as among the 10 categories of the MNIST images we considered above . As with ordinary Neural Networks and as the name implies, each neuron in this layer will be connected to all the numbers in the previous volume. +\end{itemize} + +\noindent +% !split +\subsection{Transforming images} + +CNNs transform the original image layer by layer from the original +pixel values to the final class scores. + +Observe that some layers contain +parameters and other don’t. In particular, the CNN layers perform +transformations that are a function of not only the activations in the +input volume, but also of the parameters (the weights and biases of +the neurons). On the other hand, the RELU/POOL layers will implement a +fixed function. The parameters in the CONV/FC layers will be trained +with gradient descent so that the class scores that the CNN computes +are consistent with the labels in the training set for each image. + + +% !split +\subsection{CNNs in brief} + +In summary: + +\begin{itemize} +\item A CNN architecture is in the simplest case a list of Layers that transform the image volume into an output volume (e.g.~holding the class scores) + +\item There are a few distinct types of Layers (e.g.~CONV/FC/RELU/POOL are by far the most popular) + +\item Each Layer accepts an input 3D volume and transforms it to an output 3D volume through a differentiable function + +\item Each Layer may or may not have parameters (e.g.~CONV/FC do, RELU/POOL don’t) + +\item Each Layer may or may not have additional hyperparameters (e.g.~CONV/FC/POOL do, RELU doesn’t) +\end{itemize} + +\noindent +For more material on convolutional networks, we strongly recommend +the course +\href{{https://www.uio.no/studier/emner/matnat/ifi/IN5400/index-eng.html}}{IN5400 – Machine Learning for Image Analysis} +and the slides of \href{{http://cs231n.github.io/convolutional-networks/}}{CS231} which is taught at Stanford University (consistently ranked as one of the top computer science programs in the world). \href{{http://neuralnetworksanddeeplearning.com/chap6.html}}{Michael Nielsen's book is a must read, in particular chapter 6 which deals with CNNs}. + + +% !split +\subsection{CNNs in more detail, building convolutional neural networks in Tensorflow and Keras} + + +As discussed above, CNNs are neural networks built from the assumption that the inputs +to the network are 2D images. This is important because the number of features or pixels in images +grows very fast with the image size, and an enormous number of weights and biases are needed in order to build an accurate network. + +As before, we still have our input, a hidden layer and an output. What's novel about convolutional networks +are the \textbf{convolutional} and \textbf{pooling} layers stacked in pairs between the input and the hidden layer. +In addition, the data is no longer represented as a 2D feature matrix, instead each input is a number of 2D +matrices, typically 1 for each color dimension (Red, Green, Blue). + + +% !split +\subsection{Setting it up} + +It means that to represent the entire +dataset of images, we require a 4D matrix or \textbf{tensor}. This tensor has the dimensions: +\[ +(n_{inputs},\, n_{pixels, width},\, n_{pixels, height},\, depth) . +\] + +% !split +\subsection{The MNIST dataset again} + +The MNIST dataset consists of grayscale images with a pixel size of +$28\times 28$, meaning we require $28 \times 28 = 724$ weights to each +neuron in the first hidden layer. + +If we were to analyze images of size $128\times 128$ we would require +$128 \times 128 = 16384$ weights to each neuron. Even worse if we were +dealing with color images, as most images are, we have an image matrix +of size $128\times 128$ for each color dimension (Red, Green, Blue), +meaning 3 times the number of weights $= 49152$ are required for every +single neuron in the first hidden layer. + + +% !split +\subsection{Strong correlations} +Images typically have strong local correlations, meaning that a small +part of the image varies little from its neighboring regions. If for +example we have an image of a blue car, we can roughly assume that a +small blue part of the image is surrounded by other blue regions. + +Therefore, instead of connecting every single pixel to a neuron in the +first hidden layer, as we have previously done with deep neural +networks, we can instead connect each neuron to a small part of the +image (in all 3 RGB depth dimensions). The size of each small area is +fixed, and known as a \href{{https://en.wikipedia.org/wiki/Receptive_field}}{receptive}. + + +% !split +\subsection{Layers of a CNN} +The layers of a convolutional neural network arrange neurons in 3D: width, height and depth. +The input image is typically a square matrix of depth 3. + +A \textbf{convolution} is performed on the image which outputs +a 3D volume of neurons. The weights to the input are arranged in a number of 2D matrices, known as \textbf{filters}. + + +Each filter slides along the input image, taking the dot product +between each small part of the image and the filter, in all depth +dimensions. This is then passed through a non-linear function, +typically the \textbf{Rectified Linear (ReLu)} function, which serves as the +activation of the neurons in the first convolutional layer. This is +further passed through a \textbf{pooling layer}, which reduces the size of the +convolutional layer, e.g.~by taking the maximum or average across some +small regions, and this serves as input to the next convolutional +layer. + + +% !split +\subsection{Systematic reduction} + +By systematically reducing the size of the input volume, through +convolution and pooling, the network should create representations of +small parts of the input, and then from them assemble representations +of larger areas. The final pooling layer is flattened to serve as +input to a hidden layer, such that each neuron in the final pooling +layer is connected to every single neuron in the hidden layer. This +then serves as input to the output layer, e.g.~a softmax output for +classification. + + +% !split +\subsection{Prerequisites: Collect and pre-process data} +\bpycod +# import necessary packages +import numpy as np +import matplotlib.pyplot as plt +from sklearn import datasets + + +# ensure the same random numbers appear every time +np.random.seed(0) + +# display images in notebook +%matplotlib inline +plt.rcParams['figure.figsize'] = (12,12) + + +# download MNIST dataset +digits = datasets.load_digits() + +# define inputs and labels +inputs = digits.images +labels = digits.target + +# RGB images have a depth of 3 +# our images are grayscale so they should have a depth of 1 +inputs = inputs[:,:,:,np.newaxis] + +print("inputs = (n_inputs, pixel_width, pixel_height, depth) = " + str(inputs.shape)) +print("labels = (n_inputs) = " + str(labels.shape)) + + +# choose some random images to display +n_inputs = len(inputs) +indices = np.arange(n_inputs) +random_indices = np.random.choice(indices, size=5) + +for i, image in enumerate(digits.images[random_indices]): + plt.subplot(1, 5, i+1) + plt.axis('off') + plt.imshow(image, cmap=plt.cm.gray_r, interpolation='nearest') + plt.title("Label: %d" % digits.target[random_indices[i]]) +plt.show() +\epycod + + +% !split +\subsection{Importing Keras and Tensorflow} +\bpycod +from keras.utils import to_categorical +from sklearn.model_selection import train_test_split + +# representation of labels +labels = to_categorical(labels) + +# split into train and test data +# one-liner from scikit-learn library +train_size = 0.8 +test_size = 1 - train_size +X_train, X_test, Y_train, Y_test = train_test_split(inputs, labels, train_size=train_size, + test_size=test_size) +\epycod + +% !split +\subsection{Using TensorFlow backend} + +We need to define model and architecture and choose cost function and optmizer. +\bpycid + +import tensorflow as tf + +class ConvolutionalNeuralNetworkTensorflow: + def __init__( + self, + X_train, + Y_train, + X_test, + Y_test, + n_filters=10, + n_neurons_connected=50, + n_categories=10, + receptive_field=3, + stride=1, + padding=1, + epochs=10, + batch_size=100, + eta=0.1, + lmbd=0.0): + + self.global_step = tf.Variable(0, dtype=tf.int32, trainable=False, name='global_step') + + self.X_train = X_train + self.Y_train = Y_train + self.X_test = X_test + self.Y_test = Y_test + + self.n_inputs, self.input_width, self.input_height, self.depth = X_train.shape + + self.n_filters = n_filters + self.n_downsampled = int(self.input_width*self.input_height*n_filters / 4) + self.n_neurons_connected = n_neurons_connected + self.n_categories = n_categories + + self.receptive_field = receptive_field + self.stride = stride + self.strides = [stride, stride, stride, stride] + self.padding = padding + + self.epochs = epochs + self.batch_size = batch_size + self.iterations = self.n_inputs // self.batch_size + self.eta = eta + self.lmbd = lmbd + + self.create_placeholders() + self.create_CNN() + self.create_loss() + self.create_optimiser() + self.create_accuracy() + + def create_placeholders(self): + with tf.name_scope('data'): + self.X = tf.placeholder(tf.float32, shape=(None, self.input_width, self.input_height, self.depth), name='X_data') + self.Y = tf.placeholder(tf.float32, shape=(None, self.n_categories), name='Y_data') + + def create_CNN(self): + with tf.name_scope('CNN'): + + # Convolutional layer + self.W_conv = self.weight_variable([self.receptive_field, self.receptive_field, self.depth, self.n_filters], name='conv', dtype=tf.float32) + b_conv = self.weight_variable([self.n_filters], name='conv', dtype=tf.float32) + z_conv = tf.nn.conv2d(self.X, self.W_conv, self.strides, padding='SAME', name='conv') + b_conv + a_conv = tf.nn.relu(z_conv) + + # 2x2 max pooling + a_pool = tf.nn.max_pool(a_conv, [1, 2, 2, 1], [1, 2, 2, 1], padding='SAME', name='pool') + + # Fully connected layer + a_pool_flat = tf.reshape(a_pool, [-1, self.n_downsampled]) + self.W_fc = self.weight_variable([self.n_downsampled, self.n_neurons_connected], name='fc', dtype=tf.float32) + b_fc = self.bias_variable([self.n_neurons_connected], name='fc', dtype=tf.float32) + a_fc = tf.nn.relu(tf.matmul(a_pool_flat, self.W_fc) + b_fc) + + # Output layer + self.W_out = self.weight_variable([self.n_neurons_connected, self.n_categories], name='out', dtype=tf.float32) + b_out = self.bias_variable([self.n_categories], name='out', dtype=tf.float32) + self.z_out = tf.matmul(a_fc, self.W_out) + b_out + + def create_loss(self): + with tf.name_scope('loss'): + softmax_loss = tf.reduce_mean(tf.nn.softmax_cross_entropy_with_logits_v2(labels=self.Y, logits=self.z_out)) + + regularizer_loss_conv = tf.nn.l2_loss(self.W_conv) + regularizer_loss_fc = tf.nn.l2_loss(self.W_fc) + regularizer_loss_out = tf.nn.l2_loss(self.W_out) + regularizer_loss = self.lmbd*(regularizer_loss_conv + regularizer_loss_fc + regularizer_loss_out) + + self.loss = softmax_loss + regularizer_loss + + def create_accuracy(self): + with tf.name_scope('accuracy'): + probabilities = tf.nn.softmax(self.z_out) + predictions = tf.argmax(probabilities, 1) + labels = tf.argmax(self.Y, 1) + + correct_predictions = tf.equal(predictions, labels) + correct_predictions = tf.cast(correct_predictions, tf.float32) + self.accuracy = tf.reduce_mean(correct_predictions) + + def create_optimiser(self): + with tf.name_scope('optimizer'): + self.optimizer = tf.train.GradientDescentOptimizer(learning_rate=self.eta).minimize(self.loss, global_step=self.global_step) + + def weight_variable(self, shape, name='', dtype=tf.float32): + initial = tf.truncated_normal(shape, stddev=0.1) + return tf.Variable(initial, name=name, dtype=dtype) + + def bias_variable(self, shape, name='', dtype=tf.float32): + initial = tf.constant(0.1, shape=shape) + return tf.Variable(initial, name=name, dtype=dtype) + + def fit(self): + data_indices = np.arange(self.n_inputs) + + with tf.Session() as sess: + sess.run(tf.global_variables_initializer()) + for i in range(self.epochs): + for j in range(self.iterations): + chosen_datapoints = np.random.choice(data_indices, size=self.batch_size, replace=False) + batch_X, batch_Y = self.X_train[chosen_datapoints], self.Y_train[chosen_datapoints] + + sess.run([CNN.loss, CNN.optimizer], + feed_dict={CNN.X: batch_X, + CNN.Y: batch_Y}) + accuracy = sess.run(CNN.accuracy, + feed_dict={CNN.X: batch_X, + CNN.Y: batch_Y}) + step = sess.run(CNN.global_step) + + self.train_loss, self.train_accuracy = sess.run([CNN.loss, CNN.accuracy], + feed_dict={CNN.X: self.X_train, + CNN.Y: self.Y_train}) + + self.test_loss, self.test_accuracy = sess.run([CNN.loss, CNN.accuracy], + feed_dict={CNN.X: self.X_test, + CNN.Y: self.Y_test}) +\epycid + +% !split +\subsection{Train the model} + +We need now to train the model, evaluate it and test its performance on test data, and eventually include hyperparameters. +\bpycod +epochs = 100 +batch_size = 100 +n_filters = 10 +n_neurons_connected = 50 +n_categories = 10 + +eta_vals = np.logspace(-5, 1, 7) +lmbd_vals = np.logspace(-5, 1, 7) +CNN_tf = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object) + +for i, eta in enumerate(eta_vals): + for j, lmbd in enumerate(lmbd_vals): + CNN = ConvolutionalNeuralNetworkTensorflow(X_train, Y_train, X_test, Y_test, + n_filters=n_filters, n_neurons_connected=n_neurons_connected, + n_categories=n_categories, epochs=epochs, batch_size=batch_size, + eta=eta, lmbd=lmbd) + CNN.fit() + + print("Learning rate = ", eta) + print("Lambda = ", lmbd) + print("Test accuracy: %.3f" % CNN.test_accuracy) + print() + + CNN_tf[i][j] = CNN +\epycod + +% !split +\subsection{Visualizing the results} + +\bpycod +# visual representation of grid search +# uses seaborn heatmap, could probably do this in matplotlib +import seaborn as sns + +sns.set() + +train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals))) +test_accuracy = np.zeros((len(eta_vals), len(lmbd_vals))) + +for i in range(len(eta_vals)): + for j in range(len(lmbd_vals)): + CNN = CNN_tf[i][j] + + train_accuracy[i][j] = CNN.train_accuracy + test_accuracy[i][j] = CNN.test_accuracy + + +fig, ax = plt.subplots(figsize = (10, 10)) +sns.heatmap(train_accuracy, annot=True, ax=ax, cmap="viridis") +ax.set_title("Training Accuracy") +ax.set_ylabel("$\eta$") +ax.set_xlabel("$\lambda$") +plt.show() + +fig, ax = plt.subplots(figsize = (10, 10)) +sns.heatmap(test_accuracy, annot=True, ax=ax, cmap="viridis") +ax.set_title("Test Accuracy") +ax.set_ylabel("$\eta$") +ax.set_xlabel("$\lambda$") +plt.show() +\epycod + +% !split +\subsection{Running with Keras} + +\bpycod +from keras.models import Sequential +from keras.layers.convolutional import Conv2D +from keras.layers.convolutional import MaxPooling2D +from keras.layers import Flatten +from keras.layers import Dense +from keras.regularizers import l2 +from keras.optimizers import SGD + +def create_convolutional_neural_network_keras(input_shape, receptive_field, + n_filters, n_neurons_connected, n_categories, + eta, lmbd): + model = Sequential() + model.add(Conv2D(n_filters, (receptive_field, receptive_field), input_shape=input_shape, padding='same', + activation='relu', kernel_regularizer=l2(lmbd))) + model.add(MaxPooling2D(pool_size=(2, 2))) + model.add(Flatten()) + model.add(Dense(n_neurons_connected, activation='relu', kernel_regularizer=l2(lmbd))) + model.add(Dense(n_categories, activation='softmax', kernel_regularizer=l2(lmbd))) + + sgd = SGD(lr=eta) + model.compile(loss='categorical_crossentropy', optimizer=sgd, metrics=['accuracy']) + + return model + +epochs = 100 +batch_size = 100 +input_shape = X_train.shape[1:4] +receptive_field = 3 +n_filters = 10 +n_neurons_connected = 50 +n_categories = 10 + +eta_vals = np.logspace(-5, 1, 7) +lmbd_vals = np.logspace(-5, 1, 7) +\epycod + +% !split +\subsection{Final part} + +\bpycod +CNN_keras = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object) + +for i, eta in enumerate(eta_vals): + for j, lmbd in enumerate(lmbd_vals): + CNN = create_convolutional_neural_network_keras(input_shape, receptive_field, + n_filters, n_neurons_connected, n_categories, + eta, lmbd) + CNN.fit(X_train, Y_train, epochs=epochs, batch_size=batch_size, verbose=0) + scores = CNN.evaluate(X_test, Y_test) + + CNN_keras[i][j] = CNN + + print("Learning rate = ", eta) + print("Lambda = ", lmbd) + print("Test accuracy: %.3f" % scores[1]) + print() +\epycod + +% !split +\subsection{Final visualization} + +\bccq +# visual representation of grid search +# uses seaborn heatmap, could probably do this in matplotlib +import seaborn as sns + +sns.set() + +train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals))) +test_accuracy = np.zeros((len(eta_vals), len(lmbd_vals))) + +for i in range(len(eta_vals)): + for j in range(len(lmbd_vals)): + CNN = CNN_keras[i][j] + + train_accuracy[i][j] = CNN.evaluate(X_train, Y_train)[1] + test_accuracy[i][j] = CNN.evaluate(X_test, Y_test)[1] + + +fig, ax = plt.subplots(figsize = (10, 10)) +sns.heatmap(train_accuracy, annot=True, ax=ax, cmap="viridis") +ax.set_title("Training Accuracy") +ax.set_ylabel("$\eta$") +ax.set_xlabel("$\lambda$") +plt.show() + +fig, ax = plt.subplots(figsize = (10, 10)) +sns.heatmap(test_accuracy, annot=True, ax=ax, cmap="viridis") +ax.set_title("Test Accuracy") +ax.set_ylabel("$\eta$") +ax.set_xlabel("$\lambda$") +plt.show() +\eccq + +% !split +\subsection{Fun links} + +\begin{enumerate} +\item \href{{https://arxiv.org/abs/1604.07316}}{Self-Driving cars using a convolutional neural network} + +\item \href{{https://deepdreamgenerator.com/}}{Abstract art using convolutional neural networks} +\end{enumerate} + +\noindent + +% ------------------- end of main content --------------- + +% #ifdef PREAMBLE +\end{document} +% #endif + diff --git a/doc/src/CNN/cnn.tex b/doc/src/CNN/cnn.tex new file mode 100644 index 000000000..6fb87ff00 --- /dev/null +++ b/doc/src/CNN/cnn.tex @@ -0,0 +1,781 @@ +%% +%% Automatically generated file from DocOnce source +%% (https://github.com/hplgit/doconce/) +%% +%% + + +%-------------------- begin preamble ---------------------- + +\documentclass[% +oneside, % oneside: electronic viewing, twoside: printing +final, % draft: marks overfull hboxes, figures with paths +10pt]{article} + +\listfiles % print all files needed to compile this document + +\usepackage{relsize,makeidx,color,setspace,amsmath,amsfonts,amssymb} +\usepackage[table]{xcolor} +\usepackage{bm,ltablex,microtype} + +\usepackage[pdftex]{graphicx} + +\usepackage{fancyvrb} % packages needed for verbatim environments +\usepackage{minted} +\usemintedstyle{default} + +\usepackage[T1]{fontenc} +%\usepackage[latin1]{inputenc} +\usepackage{ucs} +\usepackage[utf8x]{inputenc} + +\usepackage{lmodern} % Latin Modern fonts derived from Computer Modern + +% Hyperlinks in PDF: +\definecolor{linkcolor}{rgb}{0,0,0.4} +\usepackage{hyperref} +\hypersetup{ + breaklinks=true, + colorlinks=true, + linkcolor=linkcolor, + urlcolor=linkcolor, + citecolor=black, + filecolor=black, + %filecolor=blue, + pdfmenubar=true, + pdftoolbar=true, + bookmarksdepth=3 % Uncomment (and tweak) for PDF bookmarks with more levels than the TOC + } +%\hyperbaseurl{} % hyperlinks are relative to this root + +\setcounter{tocdepth}{2} % levels in table of contents + +% Tricks for having figures close to where they are defined: +% 1. define less restrictive rules for where to put figures +\setcounter{topnumber}{2} +\setcounter{bottomnumber}{2} +\setcounter{totalnumber}{4} +\renewcommand{\topfraction}{0.95} +\renewcommand{\bottomfraction}{0.95} +\renewcommand{\textfraction}{0} +\renewcommand{\floatpagefraction}{0.75} +% floatpagefraction must always be less than topfraction! +% 2. ensure all figures are flushed before next section +\usepackage[section]{placeins} +% 3. enable begin{figure}[H] (often leads to ugly pagebreaks) +%\usepackage{float}\restylefloat{figure} + +% --- fancyhdr package for fancy headers --- +\usepackage{fancyhdr} +\fancyhf{} % sets both header and footer to nothing +\renewcommand{\headrulewidth}{0pt} +\fancyfoot[LE,RO]{\thepage} +% Ensure copyright on titlepage (article style) and chapter pages (book style) +\fancypagestyle{plain}{ + \fancyhf{} + \fancyfoot[C]{{\footnotesize \copyright\ 1999-2019, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license}} +% \renewcommand{\footrulewidth}{0mm} + \renewcommand{\headrulewidth}{0mm} +} +% Ensure copyright on titlepages with \thispagestyle{empty} +\fancypagestyle{empty}{ + \fancyhf{} + \fancyfoot[C]{{\footnotesize \copyright\ 1999-2019, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license}} + \renewcommand{\footrulewidth}{0mm} + \renewcommand{\headrulewidth}{0mm} +} + +\pagestyle{fancy} + + +% prevent orhpans and widows +\clubpenalty = 10000 +\widowpenalty = 10000 + +% --- end of standard preamble for documents --- + + +% insert custom LaTeX commands... + +\raggedbottom +\makeindex +\usepackage[totoc]{idxlayout} % for index in the toc +\usepackage[nottoc]{tocbibind} % for references/bibliography in the toc + +%-------------------- end preamble ---------------------- + +\begin{document} + +% matching end for #ifdef PREAMBLE + +\newcommand{\exercisesection}[1]{\subsection*{#1}} + + +% ------------------- main content ---------------------- + + + +% ----------------- title ------------------------- + +\thispagestyle{empty} + +\begin{center} +{\LARGE\bf +\begin{spacing}{1.25} +Convolutional Neural Networks +\end{spacing} +} +\end{center} + +% ----------------- author(s) ------------------------- + +\begin{center} +{\bf Morten Hjorth-Jensen${}^{1, 2}$} \\ [0mm] +\end{center} + +\begin{center} +% List of all institutions: +\centerline{{\small ${}^1$Department of Physics, University of Oslo}} +\centerline{{\small ${}^2$Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University}} +\end{center} + +% ----------------- end author(s) ------------------------- + +% --- begin date --- +\begin{center} +Oct 3, 2019 +\end{center} +% --- end date --- + +\vspace{1cm} + + +% !split +\subsection*{Convolutional Neural Networks (recognizing images)} + +Convolutional Neural Networks (CNN) are very similar to ordinary Neural Networks. + +They are made up of neurons that have learnable weights and +biases. Each neuron receives some inputs, performs a dot product and +optionally follows it with a non-linearity. The whole network still +expresses a single differentiable score function: from the raw image +pixels on one end to class scores at the other. And they still have a +loss function (for example Softmax) on the last (fully-connected) layer +and all the tips/tricks we developed for learning regular Neural +Networks still apply (back propagation, gradient descent etc etc). + +What is the difference? \textbf{CNN architectures make the explicit assumption that +the inputs are images, which allows us to encode certain properties +into the architecture. These then make the forward function more +efficient to implement and vastly reduce the amount of parameters in +the network.} + +Here we provide only a superficial overview, for the more interested, we recommend highly the course +\href{{https://www.uio.no/studier/emner/matnat/ifi/IN5400/index-eng.html}}{IN5400 – Machine Learning for Image Analysis} +and the slides of \href{{http://cs231n.github.io/convolutional-networks/}}{CS231}. + +% !split +\subsection*{Regular NNs don’t scale well to full images} + +As an example, consider +an image of size $32\times 32\times 3$ (32 wide, 32 high, 3 color channels), so a +single fully-connected neuron in a first hidden layer of a regular +Neural Network would have $32\times 32\times 3 = 3072$ weights. This amount still +seems manageable, but clearly this fully-connected structure does not +scale to larger images. For example, an image of more respectable +size, say $200\times 200\times 3$, would lead to neurons that have +$200\times 200\times 3 = 120,000$ weights. + +We could have +several such neurons, and the parameters would add up quickly! Clearly, +this full connectivity is wasteful and the huge number of parameters +would quickly lead to possible overfitting. + + +\begin{figure}[!ht] % + \centerline{\includegraphics[width=0.6\linewidth]{figslides/nn.jpeg}} + \caption{ + A regular 3-layer Neural Network. + } +\end{figure} +%\clearpage % flush figures + + +% !split +\subsection*{3D volumes of neurons} + +Convolutional Neural Networks take advantage of the fact that the +input consists of images and they constrain the architecture in a more +sensible way. + +In particular, unlike a regular Neural Network, the +layers of a CNN have neurons arranged in 3 dimensions: width, +height, depth. (Note that the word depth here refers to the third +dimension of an activation volume, not to the depth of a full Neural +Network, which can refer to the total number of layers in a network.) + +To understand it better, the above example of an image +with an input volume of +activations has dimensions $32\times 32\times 3$ (width, height, +depth respectively). + +The neurons in a layer will +only be connected to a small region of the layer before it, instead of +all of the neurons in a fully-connected manner. Moreover, the final +output layer could for this specific image have dimensions $1\times 1 \times 10$, +because by the +end of the CNN architecture we will reduce the full image into a +single vector of class scores, arranged along the depth +dimension. + + +\begin{figure}[!ht] % + \centerline{\includegraphics[width=0.6\linewidth]{figslides/cnn.jpeg}} + \caption{ + A CNN arranges its neurons in three dimensions (width, height, depth), as visualized in one of the layers. Every layer of a CNN transforms the 3D input volume to a 3D output volume of neuron activations. In this example, the red input layer holds the image, so its width and height would be the dimensions of the image, and the depth would be 3 (Red, Green, Blue channels). + } +\end{figure} +%\clearpage % flush figures + + + + +% !split +\subsection*{Layers used to build CNNs} + + +A simple CNN is a sequence of layers, and every layer of a CNN +transforms one volume of activations to another through a +differentiable function. We use three main types of layers to build +CNN architectures: Convolutional Layer, Pooling Layer, and +Fully-Connected Layer (exactly as seen in regular Neural Networks). We +will stack these layers to form a full CNN architecture. + +A simple CNN for image classification could have the architecture: + +\begin{itemize} +\item \textbf{INPUT} ($32\times 32 \times 3$) will hold the raw pixel values of the image, in this case an image of width 32, height 32, and with three color channels R,G,B. + +\item \textbf{CONV} (convolutional )layer will compute the output of neurons that are connected to local regions in the input, each computing a dot product between their weights and a small region they are connected to in the input volume. This may result in volume such as $[32\times 32\times 12]$ if we decided to use 12 filters. + +\item \textbf{RELU} layer will apply an elementwise activation function, such as the $max(0,x)$ thresholding at zero. This leaves the size of the volume unchanged ($[32\times 32\times 12]$). + +\item \textbf{POOL} (pooling) layer will perform a downsampling operation along the spatial dimensions (width, height), resulting in volume such as $[16\times 16\times 12]$. + +\item \textbf{FC} (i.e.~fully-connected) layer will compute the class scores, resulting in volume of size $[1\times 1\times 10]$, where each of the 10 numbers correspond to a class score, such as among the 10 categories of the MNIST images we considered above . As with ordinary Neural Networks and as the name implies, each neuron in this layer will be connected to all the numbers in the previous volume. +\end{itemize} + +\noindent +% !split +\subsection*{Transforming images} + +CNNs transform the original image layer by layer from the original +pixel values to the final class scores. + +Observe that some layers contain +parameters and other don’t. In particular, the CNN layers perform +transformations that are a function of not only the activations in the +input volume, but also of the parameters (the weights and biases of +the neurons). On the other hand, the RELU/POOL layers will implement a +fixed function. The parameters in the CONV/FC layers will be trained +with gradient descent so that the class scores that the CNN computes +are consistent with the labels in the training set for each image. + + +% !split +\subsection*{CNNs in brief} + +In summary: + +\begin{itemize} +\item A CNN architecture is in the simplest case a list of Layers that transform the image volume into an output volume (e.g.~holding the class scores) + +\item There are a few distinct types of Layers (e.g.~CONV/FC/RELU/POOL are by far the most popular) + +\item Each Layer accepts an input 3D volume and transforms it to an output 3D volume through a differentiable function + +\item Each Layer may or may not have parameters (e.g.~CONV/FC do, RELU/POOL don’t) + +\item Each Layer may or may not have additional hyperparameters (e.g.~CONV/FC/POOL do, RELU doesn’t) +\end{itemize} + +\noindent +For more material on convolutional networks, we strongly recommend +the course +\href{{https://www.uio.no/studier/emner/matnat/ifi/IN5400/index-eng.html}}{IN5400 – Machine Learning for Image Analysis} +and the slides of \href{{http://cs231n.github.io/convolutional-networks/}}{CS231} which is taught at Stanford University (consistently ranked as one of the top computer science programs in the world). \href{{http://neuralnetworksanddeeplearning.com/chap6.html}}{Michael Nielsen's book is a must read, in particular chapter 6 which deals with CNNs}. + + +% !split +\subsection*{CNNs in more detail, building convolutional neural networks in Tensorflow and Keras} + + +As discussed above, CNNs are neural networks built from the assumption that the inputs +to the network are 2D images. This is important because the number of features or pixels in images +grows very fast with the image size, and an enormous number of weights and biases are needed in order to build an accurate network. + +As before, we still have our input, a hidden layer and an output. What's novel about convolutional networks +are the \textbf{convolutional} and \textbf{pooling} layers stacked in pairs between the input and the hidden layer. +In addition, the data is no longer represented as a 2D feature matrix, instead each input is a number of 2D +matrices, typically 1 for each color dimension (Red, Green, Blue). + + +% !split +\subsection*{Setting it up} + +It means that to represent the entire +dataset of images, we require a 4D matrix or \textbf{tensor}. This tensor has the dimensions: +\[ +(n_{inputs},\, n_{pixels, width},\, n_{pixels, height},\, depth) . +\] + +% !split +\subsection*{The MNIST dataset again} + +The MNIST dataset consists of grayscale images with a pixel size of +$28\times 28$, meaning we require $28 \times 28 = 724$ weights to each +neuron in the first hidden layer. + +If we were to analyze images of size $128\times 128$ we would require +$128 \times 128 = 16384$ weights to each neuron. Even worse if we were +dealing with color images, as most images are, we have an image matrix +of size $128\times 128$ for each color dimension (Red, Green, Blue), +meaning 3 times the number of weights $= 49152$ are required for every +single neuron in the first hidden layer. + + +% !split +\subsection*{Strong correlations} +Images typically have strong local correlations, meaning that a small +part of the image varies little from its neighboring regions. If for +example we have an image of a blue car, we can roughly assume that a +small blue part of the image is surrounded by other blue regions. + +Therefore, instead of connecting every single pixel to a neuron in the +first hidden layer, as we have previously done with deep neural +networks, we can instead connect each neuron to a small part of the +image (in all 3 RGB depth dimensions). The size of each small area is +fixed, and known as a \href{{https://en.wikipedia.org/wiki/Receptive_field}}{receptive}. + + +% !split +\subsection*{Layers of a CNN} +The layers of a convolutional neural network arrange neurons in 3D: width, height and depth. +The input image is typically a square matrix of depth 3. + +A \textbf{convolution} is performed on the image which outputs +a 3D volume of neurons. The weights to the input are arranged in a number of 2D matrices, known as \textbf{filters}. + + +Each filter slides along the input image, taking the dot product +between each small part of the image and the filter, in all depth +dimensions. This is then passed through a non-linear function, +typically the \textbf{Rectified Linear (ReLu)} function, which serves as the +activation of the neurons in the first convolutional layer. This is +further passed through a \textbf{pooling layer}, which reduces the size of the +convolutional layer, e.g.~by taking the maximum or average across some +small regions, and this serves as input to the next convolutional +layer. + + +% !split +\subsection*{Systematic reduction} + +By systematically reducing the size of the input volume, through +convolution and pooling, the network should create representations of +small parts of the input, and then from them assemble representations +of larger areas. The final pooling layer is flattened to serve as +input to a hidden layer, such that each neuron in the final pooling +layer is connected to every single neuron in the hidden layer. This +then serves as input to the output layer, e.g.~a softmax output for +classification. + + +% !split +\subsection*{Prerequisites: Collect and pre-process data} +\begin{minted}[fontsize=\fontsize{9pt}{9pt},linenos=false,mathescape,baselinestretch=1.0,fontfamily=tt,xleftmargin=7mm]{python} +# import necessary packages +import numpy as np +import matplotlib.pyplot as plt +from sklearn import datasets + + +# ensure the same random numbers appear every time +np.random.seed(0) + +# display images in notebook +%matplotlib inline +plt.rcParams['figure.figsize'] = (12,12) + + +# download MNIST dataset +digits = datasets.load_digits() + +# define inputs and labels +inputs = digits.images +labels = digits.target + +# RGB images have a depth of 3 +# our images are grayscale so they should have a depth of 1 +inputs = inputs[:,:,:,np.newaxis] + +print("inputs = (n_inputs, pixel_width, pixel_height, depth) = " + str(inputs.shape)) +print("labels = (n_inputs) = " + str(labels.shape)) + + +# choose some random images to display +n_inputs = len(inputs) +indices = np.arange(n_inputs) +random_indices = np.random.choice(indices, size=5) + +for i, image in enumerate(digits.images[random_indices]): + plt.subplot(1, 5, i+1) + plt.axis('off') + plt.imshow(image, cmap=plt.cm.gray_r, interpolation='nearest') + plt.title("Label: %d" % digits.target[random_indices[i]]) +plt.show() +\end{minted} + + +% !split +\subsection*{Importing Keras and Tensorflow} +\begin{minted}[fontsize=\fontsize{9pt}{9pt},linenos=false,mathescape,baselinestretch=1.0,fontfamily=tt,xleftmargin=7mm]{python} +from keras.utils import to_categorical +from sklearn.model_selection import train_test_split + +# representation of labels +labels = to_categorical(labels) + +# split into train and test data +# one-liner from scikit-learn library +train_size = 0.8 +test_size = 1 - train_size +X_train, X_test, Y_train, Y_test = train_test_split(inputs, labels, train_size=train_size, + test_size=test_size) +\end{minted} + +% !split +\subsection*{Using TensorFlow backend} + +We need to define model and architecture and choose cost function and optmizer. +\begin{Verbatim}[numbers=none,fontsize=\fontsize{9pt}{9pt},baselinestretch=0.95] + +import tensorflow as tf + +class ConvolutionalNeuralNetworkTensorflow: + def __init__( + self, + X_train, + Y_train, + X_test, + Y_test, + n_filters=10, + n_neurons_connected=50, + n_categories=10, + receptive_field=3, + stride=1, + padding=1, + epochs=10, + batch_size=100, + eta=0.1, + lmbd=0.0): + + self.global_step = tf.Variable(0, dtype=tf.int32, trainable=False, name='global_step') + + self.X_train = X_train + self.Y_train = Y_train + self.X_test = X_test + self.Y_test = Y_test + + self.n_inputs, self.input_width, self.input_height, self.depth = X_train.shape + + self.n_filters = n_filters + self.n_downsampled = int(self.input_width*self.input_height*n_filters / 4) + self.n_neurons_connected = n_neurons_connected + self.n_categories = n_categories + + self.receptive_field = receptive_field + self.stride = stride + self.strides = [stride, stride, stride, stride] + self.padding = padding + + self.epochs = epochs + self.batch_size = batch_size + self.iterations = self.n_inputs // self.batch_size + self.eta = eta + self.lmbd = lmbd + + self.create_placeholders() + self.create_CNN() + self.create_loss() + self.create_optimiser() + self.create_accuracy() + + def create_placeholders(self): + with tf.name_scope('data'): + self.X = tf.placeholder(tf.float32, shape=(None, self.input_width, self.input_height, self.depth), name='X_data') + self.Y = tf.placeholder(tf.float32, shape=(None, self.n_categories), name='Y_data') + + def create_CNN(self): + with tf.name_scope('CNN'): + + # Convolutional layer + self.W_conv = self.weight_variable([self.receptive_field, self.receptive_field, self.depth, self.n_filters], name='conv', dtype=tf.float32) + b_conv = self.weight_variable([self.n_filters], name='conv', dtype=tf.float32) + z_conv = tf.nn.conv2d(self.X, self.W_conv, self.strides, padding='SAME', name='conv') + b_conv + a_conv = tf.nn.relu(z_conv) + + # 2x2 max pooling + a_pool = tf.nn.max_pool(a_conv, [1, 2, 2, 1], [1, 2, 2, 1], padding='SAME', name='pool') + + # Fully connected layer + a_pool_flat = tf.reshape(a_pool, [-1, self.n_downsampled]) + self.W_fc = self.weight_variable([self.n_downsampled, self.n_neurons_connected], name='fc', dtype=tf.float32) + b_fc = self.bias_variable([self.n_neurons_connected], name='fc', dtype=tf.float32) + a_fc = tf.nn.relu(tf.matmul(a_pool_flat, self.W_fc) + b_fc) + + # Output layer + self.W_out = self.weight_variable([self.n_neurons_connected, self.n_categories], name='out', dtype=tf.float32) + b_out = self.bias_variable([self.n_categories], name='out', dtype=tf.float32) + self.z_out = tf.matmul(a_fc, self.W_out) + b_out + + def create_loss(self): + with tf.name_scope('loss'): + softmax_loss = tf.reduce_mean(tf.nn.softmax_cross_entropy_with_logits_v2(labels=self.Y, logits=self.z_out)) + + regularizer_loss_conv = tf.nn.l2_loss(self.W_conv) + regularizer_loss_fc = tf.nn.l2_loss(self.W_fc) + regularizer_loss_out = tf.nn.l2_loss(self.W_out) + regularizer_loss = self.lmbd*(regularizer_loss_conv + regularizer_loss_fc + regularizer_loss_out) + + self.loss = softmax_loss + regularizer_loss + + def create_accuracy(self): + with tf.name_scope('accuracy'): + probabilities = tf.nn.softmax(self.z_out) + predictions = tf.argmax(probabilities, 1) + labels = tf.argmax(self.Y, 1) + + correct_predictions = tf.equal(predictions, labels) + correct_predictions = tf.cast(correct_predictions, tf.float32) + self.accuracy = tf.reduce_mean(correct_predictions) + + def create_optimiser(self): + with tf.name_scope('optimizer'): + self.optimizer = tf.train.GradientDescentOptimizer(learning_rate=self.eta).minimize(self.loss, global_step=self.global_step) + + def weight_variable(self, shape, name='', dtype=tf.float32): + initial = tf.truncated_normal(shape, stddev=0.1) + return tf.Variable(initial, name=name, dtype=dtype) + + def bias_variable(self, shape, name='', dtype=tf.float32): + initial = tf.constant(0.1, shape=shape) + return tf.Variable(initial, name=name, dtype=dtype) + + def fit(self): + data_indices = np.arange(self.n_inputs) + + with tf.Session() as sess: + sess.run(tf.global_variables_initializer()) + for i in range(self.epochs): + for j in range(self.iterations): + chosen_datapoints = np.random.choice(data_indices, size=self.batch_size, replace=False) + batch_X, batch_Y = self.X_train[chosen_datapoints], self.Y_train[chosen_datapoints] + + sess.run([CNN.loss, CNN.optimizer], + feed_dict={CNN.X: batch_X, + CNN.Y: batch_Y}) + accuracy = sess.run(CNN.accuracy, + feed_dict={CNN.X: batch_X, + CNN.Y: batch_Y}) + step = sess.run(CNN.global_step) + + self.train_loss, self.train_accuracy = sess.run([CNN.loss, CNN.accuracy], + feed_dict={CNN.X: self.X_train, + CNN.Y: self.Y_train}) + + self.test_loss, self.test_accuracy = sess.run([CNN.loss, CNN.accuracy], + feed_dict={CNN.X: self.X_test, + CNN.Y: self.Y_test}) +\end{Verbatim} + +% !split +\subsection*{Train the model} + +We need now to train the model, evaluate it and test its performance on test data, and eventually include hyperparameters. +\begin{minted}[fontsize=\fontsize{9pt}{9pt},linenos=false,mathescape,baselinestretch=1.0,fontfamily=tt,xleftmargin=7mm]{python} +epochs = 100 +batch_size = 100 +n_filters = 10 +n_neurons_connected = 50 +n_categories = 10 + +eta_vals = np.logspace(-5, 1, 7) +lmbd_vals = np.logspace(-5, 1, 7) +CNN_tf = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object) + +for i, eta in enumerate(eta_vals): + for j, lmbd in enumerate(lmbd_vals): + CNN = ConvolutionalNeuralNetworkTensorflow(X_train, Y_train, X_test, Y_test, + n_filters=n_filters, n_neurons_connected=n_neurons_connected, + n_categories=n_categories, epochs=epochs, batch_size=batch_size, + eta=eta, lmbd=lmbd) + CNN.fit() + + print("Learning rate = ", eta) + print("Lambda = ", lmbd) + print("Test accuracy: %.3f" % CNN.test_accuracy) + print() + + CNN_tf[i][j] = CNN +\end{minted} + +% !split +\subsection*{Visualizing the results} + +\begin{minted}[fontsize=\fontsize{9pt}{9pt},linenos=false,mathescape,baselinestretch=1.0,fontfamily=tt,xleftmargin=7mm]{python} +# visual representation of grid search +# uses seaborn heatmap, could probably do this in matplotlib +import seaborn as sns + +sns.set() + +train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals))) +test_accuracy = np.zeros((len(eta_vals), len(lmbd_vals))) + +for i in range(len(eta_vals)): + for j in range(len(lmbd_vals)): + CNN = CNN_tf[i][j] + + train_accuracy[i][j] = CNN.train_accuracy + test_accuracy[i][j] = CNN.test_accuracy + + +fig, ax = plt.subplots(figsize = (10, 10)) +sns.heatmap(train_accuracy, annot=True, ax=ax, cmap="viridis") +ax.set_title("Training Accuracy") +ax.set_ylabel("$\eta$") +ax.set_xlabel("$\lambda$") +plt.show() + +fig, ax = plt.subplots(figsize = (10, 10)) +sns.heatmap(test_accuracy, annot=True, ax=ax, cmap="viridis") +ax.set_title("Test Accuracy") +ax.set_ylabel("$\eta$") +ax.set_xlabel("$\lambda$") +plt.show() +\end{minted} + +% !split +\subsection*{Running with Keras} + +\begin{minted}[fontsize=\fontsize{9pt}{9pt},linenos=false,mathescape,baselinestretch=1.0,fontfamily=tt,xleftmargin=7mm]{python} +from keras.models import Sequential +from keras.layers.convolutional import Conv2D +from keras.layers.convolutional import MaxPooling2D +from keras.layers import Flatten +from keras.layers import Dense +from keras.regularizers import l2 +from keras.optimizers import SGD + +def create_convolutional_neural_network_keras(input_shape, receptive_field, + n_filters, n_neurons_connected, n_categories, + eta, lmbd): + model = Sequential() + model.add(Conv2D(n_filters, (receptive_field, receptive_field), input_shape=input_shape, padding='same', + activation='relu', kernel_regularizer=l2(lmbd))) + model.add(MaxPooling2D(pool_size=(2, 2))) + model.add(Flatten()) + model.add(Dense(n_neurons_connected, activation='relu', kernel_regularizer=l2(lmbd))) + model.add(Dense(n_categories, activation='softmax', kernel_regularizer=l2(lmbd))) + + sgd = SGD(lr=eta) + model.compile(loss='categorical_crossentropy', optimizer=sgd, metrics=['accuracy']) + + return model + +epochs = 100 +batch_size = 100 +input_shape = X_train.shape[1:4] +receptive_field = 3 +n_filters = 10 +n_neurons_connected = 50 +n_categories = 10 + +eta_vals = np.logspace(-5, 1, 7) +lmbd_vals = np.logspace(-5, 1, 7) +\end{minted} + +% !split +\subsection*{Final part} + +\begin{minted}[fontsize=\fontsize{9pt}{9pt},linenos=false,mathescape,baselinestretch=1.0,fontfamily=tt,xleftmargin=7mm]{python} +CNN_keras = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object) + +for i, eta in enumerate(eta_vals): + for j, lmbd in enumerate(lmbd_vals): + CNN = create_convolutional_neural_network_keras(input_shape, receptive_field, + n_filters, n_neurons_connected, n_categories, + eta, lmbd) + CNN.fit(X_train, Y_train, epochs=epochs, batch_size=batch_size, verbose=0) + scores = CNN.evaluate(X_test, Y_test) + + CNN_keras[i][j] = CNN + + print("Learning rate = ", eta) + print("Lambda = ", lmbd) + print("Test accuracy: %.3f" % scores[1]) + print() +\end{minted} + +% !split +\subsection*{Final visualization} + +\begin{Verbatim}[numbers=none,fontsize=\fontsize{9pt}{9pt},baselinestretch=0.95] +# visual representation of grid search +# uses seaborn heatmap, could probably do this in matplotlib +import seaborn as sns + +sns.set() + +train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals))) +test_accuracy = np.zeros((len(eta_vals), len(lmbd_vals))) + +for i in range(len(eta_vals)): + for j in range(len(lmbd_vals)): + CNN = CNN_keras[i][j] + + train_accuracy[i][j] = CNN.evaluate(X_train, Y_train)[1] + test_accuracy[i][j] = CNN.evaluate(X_test, Y_test)[1] + + +fig, ax = plt.subplots(figsize = (10, 10)) +sns.heatmap(train_accuracy, annot=True, ax=ax, cmap="viridis") +ax.set_title("Training Accuracy") +ax.set_ylabel("$\eta$") +ax.set_xlabel("$\lambda$") +plt.show() + +fig, ax = plt.subplots(figsize = (10, 10)) +sns.heatmap(test_accuracy, annot=True, ax=ax, cmap="viridis") +ax.set_title("Test Accuracy") +ax.set_ylabel("$\eta$") +ax.set_xlabel("$\lambda$") +plt.show() +\end{Verbatim} + +% !split +\subsection*{Fun links} + +\begin{enumerate} +\item \href{{https://arxiv.org/abs/1604.07316}}{Self-Driving cars using a convolutional neural network} + +\item \href{{https://deepdreamgenerator.com/}}{Abstract art using convolutional neural networks} +\end{enumerate} + +\noindent + +% ------------------- end of main content --------------- + +\end{document} + diff --git a/doc/src/CNN/figslides/cnn.jpeg b/doc/src/CNN/figslides/cnn.jpeg new file mode 100644 index 0000000000000000000000000000000000000000..67bf3ced745f869c2069894442b02cb5c177b4c5 GIT binary patch literal 24437 zcmeFZcU)87w>Eev3R0y>6QTm5qN37k1f)w5qze&|E+En(1cLOAfPw;oQbmN&0#ZW< zLFq+8O+b1{5JG^EO#IHwz3=_qdFQ_G%x69`f6O5}IegB}Icp_*@Aa%_Jtvf3lm+0d 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zf@GqC*pTbwFx$N$3{2afF0Qe1(R<4?w(t*xD;^PTV+5_uX{Z^G}FgF;p*qKzLy2(l|~#6bXM3Q(#$79 z11pIu*S;J4ng43=|ItKF{i|Snb@`8erGhGCS#h5%6g^<})T-Iq&E8O$y)YtCw|A(E z5W|MHFVy}JmMhnrfOqzz7lxrJA?^tbWfQ8&0~?>r8>a&50aFe1?QsSLD_{UdLJnyz z>A)&H%LzNKHXRDCEX+do)Nk`9_o7#H3_*R`ma$%KFCG+0TU&?58 literal 0 HcmV?d00001 diff --git a/doc/src/CNN/make.sh b/doc/src/CNN/make.sh new file mode 100755 index 000000000..d908b90b9 --- /dev/null +++ b/doc/src/CNN/make.sh @@ -0,0 +1,95 @@ +#!/bin/sh +set -x + +function system { + "$@" + if [ $? -ne 0 ]; then + echo "make.sh: unsuccessful command $@" + echo "abort!" + exit 1 + fi +} + +if [ $# -eq 0 ]; then +echo 'bash make.sh slides1|slides2' +exit 1 +fi + +name=$1 +rm -f *.tar.gz + +opt="--encoding=utf-8" +# Note: Makefile examples contain constructions like ${PROG} which +# looks like Mako constructions, but they are not. Use --no_mako +# to turn off Mako processing. +opt="--no_mako" + +rm -f *.aux + + +html=${name}-reveal +system doconce format html $name --pygments_html_style=perldoc --keep_pygments_html_bg --html_links_in_new_window --html_output=$html $opt +system doconce slides_html $html reveal --html_slide_theme=beige + +# Plain HTML documents + +html=${name}-solarized +system doconce format html $name --pygments_html_style=perldoc --html_style=solarized3 --html_links_in_new_window --html_output=$html $opt +system doconce split_html $html.html --method=space10 + +html=${name} +system doconce format html $name --pygments_html_style=default --html_style=bloodish --html_links_in_new_window --html_output=$html $opt +system doconce split_html $html.html --method=space10 + +# Bootstrap style +html=${name}-bs +system doconce format html $name --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=$html $opt +system doconce split_html $html.html --method=split --pagination --nav_button=bottom + +# IPython notebook +system doconce format ipynb $name $opt + + +# Ordinary plain LaTeX document +rm -f *.aux # important after beamer +system doconce format pdflatex $name --minted_latex_style=trac --latex_admon=paragraph $opt +system doconce ptex2tex $name envir=minted +# Add special packages +doconce subst "% Add user's preamble" "\g<1>\n\\usepackage{simplewick}" $name.tex +doconce replace 'section{' 'section*{' $name.tex +pdflatex -shell-escape $name +pdflatex -shell-escape $name +mv -f $name.pdf ${name}-minted.pdf +cp $name.tex ${name}-plain-minted.tex + + + +# Publish +dest=../../pub +if [ ! -d $dest/$name ]; then +mkdir $dest/$name +mkdir $dest/$name/pdf +mkdir $dest/$name/html +mkdir $dest/$name/ipynb +fi +cp ${name}*.pdf $dest/$name/pdf +cp -r ${name}*.html ._${name}*.html reveal.js $dest/$name/html + +# Figures: cannot just copy link, need to physically copy the files +if [ -d fig-${name} ]; then +if [ ! -d $dest/$name/html/fig-$name ]; then +mkdir $dest/$name/html/fig-$name +fi +cp -r fig-${name}/* $dest/$name/html/fig-$name +fi + +cp ${name}.ipynb $dest/$name/ipynb +ipynb_tarfile=ipynb-${name}-src.tar.gz +if [ ! -f ${ipynb_tarfile} ]; then +cat > README.txt < (<%= grunt.template.today("yyyy-mm-dd, HH:MM") %>)\n' + + ' * http://lab.hakim.se/reveal-js\n' + + ' * MIT licensed\n' + + ' *\n' + + ' * Copyright (C) 2014 Hakim El Hattab, http://hakim.se\n' + + ' */' + }, + + qunit: { + files: [ 'test/*.html' ] + }, + + uglify: { + options: { + banner: '<%= meta.banner %>\n' + }, + build: { + src: 'js/reveal.js', + dest: 'js/reveal.min.js' + } + }, + + cssmin: { + compress: { + files: { + 'css/reveal.min.css': [ 'css/reveal.css' ] + } + } + }, + + sass: { + main: { + files: { + 'css/theme/darkgray.css': 'css/theme/source/darkgray.scss', + 'css/theme/beigesmall.css': 'css/theme/source/beigesmall.scss', + 'css/theme/cbc.css': 'css/theme/source/cbc.scss', + 'css/theme/default.css': 'css/theme/source/default.scss', + 'css/theme/beige.css': 'css/theme/source/beige.scss', + 'css/theme/night.css': 'css/theme/source/night.scss', + 'css/theme/serif.css': 'css/theme/source/serif.scss', + 'css/theme/simple.css': 'css/theme/source/simple.scss', + 'css/theme/sky.css': 'css/theme/source/sky.scss', + 'css/theme/moon.css': 'css/theme/source/moon.scss', + 'css/theme/solarized.css': 'css/theme/source/solarized.scss', + 'css/theme/blood.css': 'css/theme/source/blood.scss' + } + } + }, + + jshint: { + options: { + curly: false, + eqeqeq: true, + immed: true, + latedef: true, + newcap: true, + noarg: true, + sub: true, + undef: true, + eqnull: true, + browser: true, + expr: true, + globals: { + head: false, + module: false, + console: false, + unescape: false + } + }, + files: [ 'Gruntfile.js', 'js/reveal.js' ] + }, + + connect: { + server: { + options: { + port: port, + base: '.' + } + } + }, + + zip: { + 'reveal-js-presentation.zip': [ + 'index.html', + 'css/**', + 'js/**', + 'lib/**', + 'images/**', + 'plugin/**' + ] + }, + + watch: { + main: { + files: [ 'Gruntfile.js', 'js/reveal.js', 'css/reveal.css' ], + tasks: 'default' + }, + theme: { + files: [ 'css/theme/source/*.scss', 'css/theme/template/*.scss' ], + tasks: 'themes' + } + } + + }); + + // Dependencies + grunt.loadNpmTasks( 'grunt-contrib-qunit' ); + grunt.loadNpmTasks( 'grunt-contrib-jshint' ); + grunt.loadNpmTasks( 'grunt-contrib-cssmin' ); + grunt.loadNpmTasks( 'grunt-contrib-uglify' ); + grunt.loadNpmTasks( 'grunt-contrib-watch' ); + grunt.loadNpmTasks( 'grunt-contrib-sass' ); + grunt.loadNpmTasks( 'grunt-contrib-connect' ); + grunt.loadNpmTasks( 'grunt-zip' ); + + // Default task + grunt.registerTask( 'default', [ 'jshint', 'cssmin', 'uglify', 'qunit' ] ); + + // Theme task + grunt.registerTask( 'themes', [ 'sass' ] ); + + // Package presentation to archive + grunt.registerTask( 'package', [ 'default', 'zip' ] ); + + // Serve presentation locally + grunt.registerTask( 'serve', [ 'connect', 'watch' ] ); + + // Run tests + grunt.registerTask( 'test', [ 'jshint', 'qunit' ] ); + +}; diff --git a/doc/src/CNN/reveal.js/LICENSE b/doc/src/CNN/reveal.js/LICENSE new file mode 100644 index 000000000..09623076f --- /dev/null +++ b/doc/src/CNN/reveal.js/LICENSE @@ -0,0 +1,19 @@ +Copyright (C) 2015 Hakim El Hattab, http://hakim.se + +Permission is hereby granted, free of charge, to any person obtaining a copy +of this software and associated documentation files (the "Software"), to deal +in the Software without restriction, including without limitation the rights +to use, copy, modify, merge, publish, distribute, sublicense, and/or sell +copies of the Software, and to permit persons to whom the Software is +furnished to do so, subject to the following conditions: + +The above copyright notice and this permission notice shall be included in +all copies or substantial portions of the Software. + +THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR +IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, +FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE +AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER +LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, +OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN +THE SOFTWARE. \ No newline at end of file diff --git a/doc/src/CNN/reveal.js/README.md b/doc/src/CNN/reveal.js/README.md new file mode 100644 index 000000000..573b19597 --- /dev/null +++ b/doc/src/CNN/reveal.js/README.md @@ -0,0 +1,1052 @@ +# reveal.js [![Build Status](https://travis-ci.org/hakimel/reveal.js.svg?branch=master)](https://travis-ci.org/hakimel/reveal.js) + +A framework for easily creating beautiful presentations using HTML. [Check out the live demo](http://lab.hakim.se/reveal-js/). + +reveal.js comes with a broad range of features including [nested slides](https://github.com/hakimel/reveal.js#markup), [Markdown contents](https://github.com/hakimel/reveal.js#markdown), [PDF export](https://github.com/hakimel/reveal.js#pdf-export), [speaker notes](https://github.com/hakimel/reveal.js#speaker-notes) and a [JavaScript API](https://github.com/hakimel/reveal.js#api). It's best viewed in a modern browser but [fallbacks](https://github.com/hakimel/reveal.js/wiki/Browser-Support) are available to make sure your presentation can still be viewed elsewhere. + + +#### More reading: +- [Installation](#installation): Step-by-step instructions for getting reveal.js running on your computer. +- [Changelog](https://github.com/hakimel/reveal.js/releases): Up-to-date version history. +- [Examples](https://github.com/hakimel/reveal.js/wiki/Example-Presentations): Presentations created with reveal.js, add your own! +- [Browser Support](https://github.com/hakimel/reveal.js/wiki/Browser-Support): Explanation of browser support and fallbacks. +- [Plugins](https://github.com/hakimel/reveal.js/wiki/Plugins,-Tools-and-Hardware): A list of plugins that can be used to extend reveal.js. + +## Online Editor + +Presentations are written using HTML or Markdown but there's also an online editor for those of you who prefer a graphical interface. Give it a try at [http://slides.com](http://slides.com). + + +## Instructions + +### Markup + +Markup hierarchy needs to be ``
    `` where the ``
    `` represents one slide and can be repeated indefinitely. If you place multiple ``
    ``'s inside of another ``
    `` they will be shown as vertical slides. The first of the vertical slides is the "root" of the others (at the top), and it will be included in the horizontal sequence. For example: + +```html +
    +
    +
    Single Horizontal Slide
    +
    +
    Vertical Slide 1
    +
    Vertical Slide 2
    +
    +
    +
    +``` + +### Markdown + +It's possible to write your slides using Markdown. To enable Markdown, add the ```data-markdown``` attribute to your ```
    ``` elements and wrap the contents in a ``` +
    +``` + +#### External Markdown + +You can write your content as a separate file and have reveal.js load it at runtime. Note the separator arguments which determine how slides are delimited in the external file. The ```data-charset``` attribute is optional and specifies which charset to use when loading the external file. + +When used locally, this feature requires that reveal.js [runs from a local web server](#full-setup). + +```html +
    +
    +``` + +#### Element Attributes + +Special syntax (in html comment) is available for adding attributes to Markdown elements. This is useful for fragments, amongst other things. + +```html +
    + +
    +``` + +#### Slide Attributes + +Special syntax (in html comment) is available for adding attributes to the slide `
    ` elements generated by your Markdown. + +```html +
    + +
    +``` + + +### Configuration + +At the end of your page you need to initialize reveal by running the following code. Note that all config values are optional and will default as specified below. + +```javascript +Reveal.initialize({ + + // Display controls in the bottom right corner + controls: true, + + // Display a presentation progress bar + progress: true, + + // Display the page number of the current slide + slideNumber: false, + + // Push each slide change to the browser history + history: false, + + // Enable keyboard shortcuts for navigation + keyboard: true, + + // Enable the slide overview mode + overview: true, + + // Vertical centering of slides + center: true, + + // Enables touch navigation on devices with touch input + touch: true, + + // Loop the presentation + loop: false, + + // Change the presentation direction to be RTL + rtl: false, + + // Turns fragments on and off globally + fragments: true, + + // Flags if the presentation is running in an embedded mode, + // i.e. contained within a limited portion of the screen + embedded: false, + + // Flags if we should show a help overlay when the questionmark + // key is pressed + help: true, + + // Number of milliseconds between automatically proceeding to the + // next slide, disabled when set to 0, this value can be overwritten + // by using a data-autoslide attribute on your slides + autoSlide: 0, + + // Stop auto-sliding after user input + autoSlideStoppable: true, + + // Enable slide navigation via mouse wheel + mouseWheel: false, + + // Hides the address bar on mobile devices + hideAddressBar: true, + + // Opens links in an iframe preview overlay + previewLinks: false, + + // Transition style + transition: 'default', // none/fade/slide/convex/concave/zoom + + // Transition speed + transitionSpeed: 'default', // default/fast/slow + + // Transition style for full page slide backgrounds + backgroundTransition: 'default', // none/fade/slide/convex/concave/zoom + + // Number of slides away from the current that are visible + viewDistance: 3, + + // Parallax background image + parallaxBackgroundImage: '', // e.g. "'https://s3.amazonaws.com/hakim-static/reveal-js/reveal-parallax-1.jpg'" + + // Parallax background size + parallaxBackgroundSize: '', // CSS syntax, e.g. "2100px 900px" + + // Amount to move parallax background (horizontal and vertical) on slide change + // Number, e.g. 100 + parallaxBackgroundHorizontal: '', + parallaxBackgroundVertical: '' + +}); +``` + + +The configuration can be updated after initialization using the ```configure``` method: + +```javascript +// Turn autoSlide off +Reveal.configure({ autoSlide: 0 }); + +// Start auto-sliding every 5s +Reveal.configure({ autoSlide: 5000 }); +``` + + +### Dependencies + +Reveal.js doesn't _rely_ on any third party scripts to work but a few optional libraries are included by default. These libraries are loaded as dependencies in the order they appear, for example: + +```javascript +Reveal.initialize({ + dependencies: [ + // Cross-browser shim that fully implements classList - https://github.com/eligrey/classList.js/ + { src: 'lib/js/classList.js', condition: function() { return !document.body.classList; } }, + + // Interpret Markdown in
    elements + { src: 'plugin/markdown/marked.js', condition: function() { return !!document.querySelector( '[data-markdown]' ); } }, + { src: 'plugin/markdown/markdown.js', condition: function() { return !!document.querySelector( '[data-markdown]' ); } }, + + // Syntax highlight for elements + { src: 'plugin/highlight/highlight.js', async: true, callback: function() { hljs.initHighlightingOnLoad(); } }, + + // Zoom in and out with Alt+click + { src: 'plugin/zoom-js/zoom.js', async: true }, + + // Speaker notes + { src: 'plugin/notes/notes.js', async: true }, + + // Remote control your reveal.js presentation using a touch device + { src: 'plugin/remotes/remotes.js', async: true }, + + // MathJax + { src: 'plugin/math/math.js', async: true } + ] +}); +``` + +You can add your own extensions using the same syntax. The following properties are available for each dependency object: +- **src**: Path to the script to load +- **async**: [optional] Flags if the script should load after reveal.js has started, defaults to false +- **callback**: [optional] Function to execute when the script has loaded +- **condition**: [optional] Function which must return true for the script to be loaded + + +### Ready Event + +A 'ready' event is fired when reveal.js has loaded all non-async dependencies and is ready to start navigating. To check if reveal.js is already 'ready' you can call `Reveal.isReady()`. + +```javascript +Reveal.addEventListener( 'ready', function( event ) { + // event.currentSlide, event.indexh, event.indexv +} ); +``` + + +### Presentation Size + +All presentations have a normal size, that is the resolution at which they are authored. The framework will automatically scale presentations uniformly based on this size to ensure that everything fits on any given display or viewport. + +See below for a list of configuration options related to sizing, including default values: + +```javascript +Reveal.initialize({ + + ... + + // The "normal" size of the presentation, aspect ratio will be preserved + // when the presentation is scaled to fit different resolutions. Can be + // specified using percentage units. + width: 960, + height: 700, + + // Factor of the display size that should remain empty around the content + margin: 0.1, + + // Bounds for smallest/largest possible scale to apply to content + minScale: 0.2, + maxScale: 1.5 + +}); +``` + + +### Auto-sliding + +Presentations can be configured to progress through slides automatically, without any user input. To enable this you will need to tell the framework how many milliseconds it should wait between slides: + +```javascript +// Slide every five seconds +Reveal.configure({ + autoSlide: 5000 +}); +``` +When this is turned on a control element will appear that enables users to pause and resume auto-sliding. Alternatively, sliding can be paused or resumed by pressing »a« on the keyboard. Sliding is paused automatically as soon as the user starts navigating. You can disable these controls by specifying ```autoSlideStoppable: false``` in your reveal.js config. + +You can also override the slide duration for individual slides and fragments by using the ```data-autoslide``` attribute: + +```html +
    +

    After 2 seconds the first fragment will be shown.

    +

    After 10 seconds the next fragment will be shown.

    +

    Now, the fragment is displayed for 2 seconds before the next slide is shown.

    +
    +``` + +Whenever the auto-slide mode is resumed or paused the ```autoslideresumed``` and ```autoslidepaused``` events are fired. + + +### Keyboard Bindings + +If you're unhappy with any of the default keyboard bindings you can override them using the ```keyboard``` config option: + +```javascript +Reveal.configure({ + keyboard: { + 13: 'next', // go to the next slide when the ENTER key is pressed + 27: function() {}, // do something custom when ESC is pressed + 32: null // don't do anything when SPACE is pressed (i.e. disable a reveal.js default binding) + } +}); +``` + +### Lazy Loading + +When working on presentation with a lot of media or iframe content it's important to load lazily. Lazy loading means that reveal.js will only load content for the few slides nearest to the current slide. The number of slides that are preloaded is determined by the `viewDistance` configuration option. + +To enable lazy loading all you need to do is change your "src" attributes to "data-src" as shown below. This is supported for image, video, audio and iframe elements. Lazy loaded iframes will also unload when the containing slide is no longer visible. + +```html +
    + + + +
    +``` + + +### API + +The ``Reveal`` object exposes a JavaScript API for controlling navigation and reading state: + +```javascript +// Navigation +Reveal.slide( indexh, indexv, indexf ); +Reveal.left(); +Reveal.right(); +Reveal.up(); +Reveal.down(); +Reveal.prev(); +Reveal.next(); +Reveal.prevFragment(); +Reveal.nextFragment(); + +// Toggle presentation states, optionally pass true/false to force on/off +Reveal.toggleOverview(); +Reveal.togglePause(); +Reveal.toggleAutoSlide(); + +// Change a config value at runtime +Reveal.configure({ controls: true }); + +// Returns the present configuration options +Reveal.getConfig(); + +// Fetch the current scale of the presentation +Reveal.getScale(); + +// Retrieves the previous and current slide elements +Reveal.getPreviousSlide(); +Reveal.getCurrentSlide(); + +Reveal.getIndices(); // { h: 0, v: 0 } } +Reveal.getProgress(); // 0-1 +Reveal.getTotalSlides(); + +// State checks +Reveal.isFirstSlide(); +Reveal.isLastSlide(); +Reveal.isOverview(); +Reveal.isPaused(); +Reveal.isAutoSliding(); +``` + +### Slide Changed Event + +A 'slidechanged' event is fired each time the slide is changed (regardless of state). The event object holds the index values of the current slide as well as a reference to the previous and current slide HTML nodes. + +Some libraries, like MathJax (see [#226](https://github.com/hakimel/reveal.js/issues/226#issuecomment-10261609)), get confused by the transforms and display states of slides. Often times, this can be fixed by calling their update or render function from this callback. + +```javascript +Reveal.addEventListener( 'slidechanged', function( event ) { + // event.previousSlide, event.currentSlide, event.indexh, event.indexv +} ); +``` + +### Presentation State + +The presentation's current state can be fetched by using the `getState` method. A state object contains all of the information required to put the presentation back as it was when `getState` was first called. Sort of like a snapshot. It's a simple object that can easily be stringified and persisted or sent over the wire. + +```javascript +Reveal.slide( 1 ); +// we're on slide 1 + +var state = Reveal.getState(); + +Reveal.slide( 3 ); +// we're on slide 3 + +Reveal.setState( state ); +// we're back on slide 1 +``` + +### Slide States + +If you set ``data-state="somestate"`` on a slide ``
    ``, "somestate" will be applied as a class on the document element when that slide is opened. This allows you to apply broad style changes to the page based on the active slide. + +Furthermore you can also listen to these changes in state via JavaScript: + +```javascript +Reveal.addEventListener( 'somestate', function() { + // TODO: Sprinkle magic +}, false ); +``` + +### Slide Backgrounds + +Slides are contained within a limited portion of the screen by default to allow them to fit any display and scale uniformly. You can apply full page backgrounds outside of the slide area by adding a ```data-background``` attribute to your ```
    ``` elements. Four different types of backgrounds are supported: color, image, video and iframe. Below are a few examples. + +```html +
    +

    All CSS color formats are supported, like rgba() or hsl().

    +
    +
    +

    This slide will have a full-size background image.

    +
    +
    +

    This background image will be sized to 100px and repeated.

    +
    +
    +

    Video. Multiple sources can be defined using a comma separated list. Video will loop when the data-background-video-loop attribute is provided.

    +
    +
    +

    Embeds a web page as a background. Note that the page won't be interactive.

    +
    +``` + +Backgrounds transition using a fade animation by default. This can be changed to a linear sliding transition by passing ```backgroundTransition: 'slide'``` to the ```Reveal.initialize()``` call. Alternatively you can set ```data-background-transition``` on any section with a background to override that specific transition. + + +### Parallax Background + +If you want to use a parallax scrolling background, set the first two config properties below when initializing reveal.js (the other two are optional). + +```javascript +Reveal.initialize({ + + // Parallax background image + parallaxBackgroundImage: '', // e.g. "https://s3.amazonaws.com/hakim-static/reveal-js/reveal-parallax-1.jpg" + + // Parallax background size + parallaxBackgroundSize: '', // CSS syntax, e.g. "2100px 900px" - currently only pixels are supported (don't use % or auto) + + // Amount of pixels to move the parallax background per slide step, + // a value of 0 disables movement along the given axis + // These are optional, if they aren't specified they'll be calculated automatically + parallaxBackgroundHorizontal: 200, + parallaxBackgroundVertical: 50 + +}); +``` + +Make sure that the background size is much bigger than screen size to allow for some scrolling. [View example](http://lab.hakim.se/reveal-js/?parallaxBackgroundImage=https%3A%2F%2Fs3.amazonaws.com%2Fhakim-static%2Freveal-js%2Freveal-parallax-1.jpg¶llaxBackgroundSize=2100px%20900px). + + + +### Slide Transitions +The global presentation transition is set using the ```transition``` config value. You can override the global transition for a specific slide by using the ```data-transition``` attribute: + +```html +
    +

    This slide will override the presentation transition and zoom!

    +
    + +
    +

    Choose from three transition speeds: default, fast or slow!

    +
    +``` + +You can also use different in and out transitions for the same slide: + +```html +
    + The train goes on … +
    +
    + and on … +
    +
    + and stops. +
    +
    + (Passengers entering and leaving) +
    +
    + And it starts again. +
    +``` + + +Note that this does not work with the page and cube transitions. + + +### Internal links + +It's easy to link between slides. The first example below targets the index of another slide whereas the second targets a slide with an ID attribute (```
    ```): + +```html +Link +Link +``` + +You can also add relative navigation links, similar to the built in reveal.js controls, by appending one of the following classes on any element. Note that each element is automatically given an ```enabled``` class when it's a valid navigation route based on the current slide. + +```html + + + + + + +``` + + +### Fragments +Fragments are used to highlight individual elements on a slide. Every element with the class ```fragment``` will be stepped through before moving on to the next slide. Here's an example: http://lab.hakim.se/reveal-js/#/fragments + +The default fragment style is to start out invisible and fade in. This style can be changed by appending a different class to the fragment: + +```html +
    +

    grow

    +

    shrink

    +

    fade-out

    +

    visible only once

    +

    blue only once

    +

    highlight-red

    +

    highlight-green

    +

    highlight-blue

    +
    +``` + +Multiple fragments can be applied to the same element sequentially by wrapping it, this will fade in the text on the first step and fade it back out on the second. + +```html +
    + + I'll fade in, then out + +
    +``` + +The display order of fragments can be controlled using the ```data-fragment-index``` attribute. + +```html +
    +

    Appears last

    +

    Appears first

    +

    Appears second

    +
    +``` + +### Fragment events + +When a slide fragment is either shown or hidden reveal.js will dispatch an event. + +Some libraries, like MathJax (see #505), get confused by the initially hidden fragment elements. Often times this can be fixed by calling their update or render function from this callback. + +```javascript +Reveal.addEventListener( 'fragmentshown', function( event ) { + // event.fragment = the fragment DOM element +} ); +Reveal.addEventListener( 'fragmenthidden', function( event ) { + // event.fragment = the fragment DOM element +} ); +``` + +### Code syntax highlighting + +By default, Reveal is configured with [highlight.js](http://softwaremaniacs.org/soft/highlight/en/) for code syntax highlighting. Below is an example with clojure code that will be syntax highlighted. When the `data-trim` attribute is present surrounding whitespace is automatically removed. + +```html +
    + 
    
+(def lazy-fib
+  (concat
+   [0 1]
+   ((fn rfib [a b]
+        (lazy-cons (+ a b) (rfib b (+ a b)))) 0 1)))
+
    +
    +``` + +### Slide number +If you would like to display the page number of the current slide you can do so using the ```slideNumber``` configuration value. + +```javascript +// Shows the slide number using default formatting +Reveal.configure({ slideNumber: true }); + +// Slide number formatting can be configured using these variables: +// h: current slide's horizontal index +// v: current slide's vertical index +// c: current slide index (flattened) +// t: total number of slides (flattened) +Reveal.configure({ slideNumber: 'c / t' }); + +``` + + +### Overview mode + +Press "Esc" or "o" keys to toggle the overview mode on and off. While you're in this mode, you can still navigate between slides, +as if you were at 1,000 feet above your presentation. The overview mode comes with a few API hooks: + +```javascript +Reveal.addEventListener( 'overviewshown', function( event ) { /* ... */ } ); +Reveal.addEventListener( 'overviewhidden', function( event ) { /* ... */ } ); + +// Toggle the overview mode programmatically +Reveal.toggleOverview(); +``` + +### Fullscreen mode +Just press »F« on your keyboard to show your presentation in fullscreen mode. Press the »ESC« key to exit fullscreen mode. + + +### Embedded media +Embedded HTML5 `