From 027fe32f90a63ea3a51be9e104b36856f92e2535 Mon Sep 17 00:00:00 2001 From: Morten Hjorth-Jensen Date: Wed, 13 Oct 2021 16:00:24 +0200 Subject: [PATCH] update week41 --- doc/pub/week41/html/week41-bs.html | 107 +++++++----- doc/pub/week41/html/week41-reveal.html | 150 ++++++++++++++++- doc/pub/week41/html/week41-solarized.html | 166 ++++++++++++++++++- doc/pub/week41/html/week41.html | 166 ++++++++++++++++++- doc/pub/week41/ipynb/ipynb-week41-src.tar.gz | Bin 87692 -> 534110 bytes doc/pub/week41/ipynb/week41.ipynb | 139 +++++++++++++++- doc/src/week41/figslides/ImageReduction.png | Bin 0 -> 449610 bytes doc/src/week41/week41.do.txt | 114 +++++++++++++ 8 files changed, 798 insertions(+), 44 deletions(-) create mode 100644 doc/src/week41/figslides/ImageReduction.png diff --git a/doc/pub/week41/html/week41-bs.html b/doc/pub/week41/html/week41-bs.html index 5f2dc230a..64647fae9 100644 --- a/doc/pub/week41/html/week41-bs.html +++ b/doc/pub/week41/html/week41-bs.html @@ -107,6 +107,19 @@ Automatically generated HTML file from DocOnce source None, 'scikit-learn-implementation'), ('Visualization', 2, None, 'visualization'), + ('Testing our code for the XOR, OR and AND gates', + 2, + None, + 'testing-our-code-for-the-xor-or-and-and-gates'), + ('The AND and XOR Gates', 2, None, 'the-and-and-xor-gates'), + ('Representing the Data Sets', + 2, + None, + 'representing-the-data-sets'), + ('Setting up the Neural Network', + 2, + None, + 'setting-up-the-neural-network'), ('Building neural networks in Tensorflow and Keras', 2, None, @@ -162,6 +175,14 @@ Automatically generated HTML file from DocOnce source 2, None, 'limitations-of-supervised-learning-with-deep-networks'), + ('Overarching Views, a personal note', + 2, + None, + 'overarching-views-a-personal-note'), + ('From a Spherical Cow to a real one', + 2, + None, + 'from-a-spherical-cow-to-a-real-one'), ('Convolutional Neural Networks (recognizing images)', 2, None, @@ -244,7 +265,7 @@ MathJax.Hub.Config({
  • Example: binary classification problem
  • The Softmax function
  • Developing a code for doing neural networks with back propagation
  • -
  • Collect and pre-process data
  • +
  • Collect and pre-process data
  • Train and test datasets
  • Define model and architecture
  • Layers
  • @@ -262,43 +283,49 @@ MathJax.Hub.Config({
  • Visualization
  • scikit-learn implementation
  • Visualization
  • -
  • Building neural networks in Tensorflow and Keras
  • -
  • Tensorflow
  • -
  • Using Keras
  • -
  • Collect and pre-process data
  • -
  • The Breast Cancer Data, now with Keras
  • -
  • Fine-tuning neural network hyperparameters
  • -
  • Hidden layers
  • -
  • Which activation function should I use?
  • -
  • Is the Logistic activation function (Sigmoid) our choice?
  • -
  • The derivative of the Logistic funtion
  • -
  • The RELU function family
  • -
  • Which activation function should we use?
  • -
  • More on activation functions, output layers
  • -
  • Batch Normalization
  • -
  • Dropout
  • -
  • Gradient Clipping
  • -
  • A very nice website on Neural Networks
  • -
  • 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
  • -
  • Running with Keras
  • -
  • Final part
  • -
  • Final visualization
  • -
  • Fun links
  • +
  • Testing our code for the XOR, OR and AND gates
  • +
  • The AND and XOR Gates
  • +
  • Representing the Data Sets
  • +
  • Setting up the Neural Network
  • +
  • Building neural networks in Tensorflow and Keras
  • +
  • Tensorflow
  • +
  • Using Keras
  • +
  • Collect and pre-process data
  • +
  • The Breast Cancer Data, now with Keras
  • +
  • Fine-tuning neural network hyperparameters
  • +
  • Hidden layers
  • +
  • Which activation function should I use?
  • +
  • Is the Logistic activation function (Sigmoid) our choice?
  • +
  • The derivative of the Logistic funtion
  • +
  • The RELU function family
  • +
  • Which activation function should we use?
  • +
  • More on activation functions, output layers
  • +
  • Batch Normalization
  • +
  • Dropout
  • +
  • Gradient Clipping
  • +
  • A very nice website on Neural Networks
  • +
  • A top-down perspective on Neural networks
  • +
  • Limitations of supervised learning with deep networks
  • +
  • Overarching Views, a personal note
  • +
  • From a Spherical Cow to a real one
  • +
  • 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
  • +
  • Running with Keras
  • +
  • Final part
  • +
  • Final visualization
  • +
  • Fun links
  • @@ -333,7 +360,7 @@ MathJax.Hub.Config({
    [2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University

    -

    Oct 11, 2021

    +

    Oct 13, 2021


    @@ -357,7 +384,7 @@ MathJax.Hub.Config({

  • 9
  • 10
  • ...
  • -
  • 64
  • +
  • 70
  • »
  • diff --git a/doc/pub/week41/html/week41-reveal.html b/doc/pub/week41/html/week41-reveal.html index e4f976ffc..5c4fc0f15 100644 --- a/doc/pub/week41/html/week41-reveal.html +++ b/doc/pub/week41/html/week41-reveal.html @@ -148,7 +148,7 @@ MathJax.Hub.Config({
    [2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University

     
    -

    Oct 11, 2021

    +

    Oct 13, 2021


    @@ -1468,6 +1468,115 @@ plt.show() +

    +

    Testing our code for the XOR, OR and AND gates

    + +

    +Last week we discussed three different types of gates, the so-called +XOR, the OR and the AND gates. Their inputs and outputs can be +summarized using the following tables, first for the OR gate with +inputs \( x_1 \) and \( x_2 \) and outputs \( y \): + +

    + + + + + + + + + + +
    \( x_1 \) \( x_2 \) \( y \)
    0 0 0
    0 1 1
    1 0 1
    1 1 1
    +

    + + +
    +

    The AND and XOR Gates

    + +

    +The AND gate is defined as + +

    + + + + + + + + + + +
    \( x_1 \) \( x_2 \) \( y \)
    0 0 0
    0 1 0
    1 0 0
    1 1 1
    +

    +And finally we have the XOR gate + +

    + + + + + + + + + + +
    \( x_1 \) \( x_2 \) \( y \)
    0 0 0
    0 1 1
    1 0 1
    1 1 0
    +

    + + +
    +

    Representing the Data Sets

    + +

    +Our design matrix is defined by the input values \( x_1 \) and \( x_2 \). Since we have four possible outputs, our design matrix reads + +

     
    +$$ +\boldsymbol{X}=\begin{bmatrix} 0 & 0 \\ + 0 & 1 \\ + 1 & 0 \\ + 1 & 1 \end{bmatrix}, +$$ +

     
    + +while the vector of outputs is \( \boldsymbol{y}^T=[0,1,1,0] \) for the XOR gate, \( \boldsymbol{y}^T=[0,0,0,1] \) for the AND gate and \( \boldsymbol{y}^T=[0,1,1,1] \) for the OR gate. +

    + + +
    +

    Setting up the Neural Network

    + +

    +We define first our design matrix and the various input vectors. + +

    + + +

    """
    +Simple code that tests XOR, OR and AND gates with linear regression
    +"""
    +
    +import numpy as np
    +# Design matrix
    +X = np.array([ [1, 0, 0], [1, 0, 1], [1, 1, 0],[1, 1, 1]],dtype=np.float64)
    +
    +# The XOR gate 
    +yXOR = np.array( [ 0, 1 ,1, 0])
    +# The OR gate 
    +yOR = np.array( [ 0, 1 ,1, 1])
    +# The AND gate 
    +yAND = np.array( [ 0, 0 ,0, 1])
    +
    +#print(f"The values of theta for the AND gate:{ThetaAND}")
    +#print(f"The linear regression prediction  for the AND gate:{X @ ThetaAND}")
    +
    +
    + +

    Building neural networks in Tensorflow and Keras

    @@ -2226,6 +2335,45 @@ Some of these remarks are particular to DNNs, others are shared by all supervise
    +
    +

    Overarching Views, a personal note

    + +

    +The author of these lecture notes has an overarching take on many of +the machine learning algorithms we discuss here. + +

    +If we wish to understand complex systems, we need to find some +effective degrees of freedom or features that we find essential, +simply in order to reduce the complexity of the systems we are +studying. This leads, in one way or the other to dimensionality +reductions. Most of the Machine Learning methods we encounter deal +with this, whether we opt for a principal component analysis, or +clustering, or convolutional neural networks, or Ridge or Lasso +regression or random forest, yes, perhaps most machine learning +methods at large. + +

    +For neural networks and our previous discussion, we have seen that we +in essence end up with matrix-matrix and matrix-vector +multiplications. In all cases, our matrices are dense ones, and the +more data we deal with the larger the dimensionalities of the matrices +and vectors. How can we reduce such dimensionalities? One possible +answer is offered by convolutional neural networks (CNN), as +discussed below. The figure here shows a typical situation of the +reduction of information in an image and is typical of what CNNs +actually end up doing. +

    + + +
    +

    From a Spherical Cow to a real one

    + +

    +



    +
    + +

    Convolutional Neural Networks (recognizing images)

    diff --git a/doc/pub/week41/html/week41-solarized.html b/doc/pub/week41/html/week41-solarized.html index 2c58ec1d8..93ae88671 100644 --- a/doc/pub/week41/html/week41-solarized.html +++ b/doc/pub/week41/html/week41-solarized.html @@ -127,6 +127,19 @@ div { text-align: justify; text-justify: inter-word; } None, 'scikit-learn-implementation'), ('Visualization', 2, None, 'visualization'), + ('Testing our code for the XOR, OR and AND gates', + 2, + None, + 'testing-our-code-for-the-xor-or-and-and-gates'), + ('The AND and XOR Gates', 2, None, 'the-and-and-xor-gates'), + ('Representing the Data Sets', + 2, + None, + 'representing-the-data-sets'), + ('Setting up the Neural Network', + 2, + None, + 'setting-up-the-neural-network'), ('Building neural networks in Tensorflow and Keras', 2, None, @@ -182,6 +195,14 @@ div { text-align: justify; text-justify: inter-word; } 2, None, 'limitations-of-supervised-learning-with-deep-networks'), + ('Overarching Views, a personal note', + 2, + None, + 'overarching-views-a-personal-note'), + ('From a Spherical Cow to a real one', + 2, + None, + 'from-a-spherical-cow-to-a-real-one'), ('Convolutional Neural Networks (recognizing images)', 2, None, @@ -260,7 +281,7 @@ MathJax.Hub.Config({
    [2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University

    -

    Oct 11, 2021

    +

    Oct 13, 2021












    @@ -1457,6 +1478,110 @@ plt.show()











    +

    Testing our code for the XOR, OR and AND gates

    + +

    +Last week we discussed three different types of gates, the so-called +XOR, the OR and the AND gates. Their inputs and outputs can be +summarized using the following tables, first for the OR gate with +inputs \( x_1 \) and \( x_2 \) and outputs \( y \): + +

    + + + + + + + + + + +
    \( x_1 \) \( x_2 \) \( y \)
    0 0 0
    0 1 1
    1 0 1
    1 1 1
    +

    +









    + +

    The AND and XOR Gates

    + +

    +The AND gate is defined as + +

    + + + + + + + + + + +
    \( x_1 \) \( x_2 \) \( y \)
    0 0 0
    0 1 0
    1 0 0
    1 1 1
    +

    +And finally we have the XOR gate + +

    + + + + + + + + + + +
    \( x_1 \) \( x_2 \) \( y \)
    0 0 0
    0 1 1
    1 0 1
    1 1 0
    +

    +









    + +

    Representing the Data Sets

    + +

    +Our design matrix is defined by the input values \( x_1 \) and \( x_2 \). Since we have four possible outputs, our design matrix reads + +$$ +\boldsymbol{X}=\begin{bmatrix} 0 & 0 \\ + 0 & 1 \\ + 1 & 0 \\ + 1 & 1 \end{bmatrix}, +$$ + +while the vector of outputs is \( \boldsymbol{y}^T=[0,1,1,0] \) for the XOR gate, \( \boldsymbol{y}^T=[0,0,0,1] \) for the AND gate and \( \boldsymbol{y}^T=[0,1,1,1] \) for the OR gate. + +

    +









    + +

    Setting up the Neural Network

    + +

    +We define first our design matrix and the various input vectors. + +

    + + +

    """
    +Simple code that tests XOR, OR and AND gates with linear regression
    +"""
    +
    +import numpy as np
    +# Design matrix
    +X = np.array([ [1, 0, 0], [1, 0, 1], [1, 1, 0],[1, 1, 1]],dtype=np.float64)
    +
    +# The XOR gate 
    +yXOR = np.array( [ 0, 1 ,1, 0])
    +# The OR gate 
    +yOR = np.array( [ 0, 1 ,1, 1])
    +# The AND gate 
    +yAND = np.array( [ 0, 0 ,0, 1])
    +
    +#print(f"The values of theta for the AND gate:{ThetaAND}")
    +#print(f"The linear regression prediction  for the AND gate:{X @ ThetaAND}")
    +
    +

    +









    +

    Building neural networks in Tensorflow and Keras

    @@ -2205,6 +2330,45 @@ Some of these remarks are particular to DNNs, others are shared by all supervise











    +

    Overarching Views, a personal note

    + +

    +The author of these lecture notes has an overarching take on many of +the machine learning algorithms we discuss here. + +

    +If we wish to understand complex systems, we need to find some +effective degrees of freedom or features that we find essential, +simply in order to reduce the complexity of the systems we are +studying. This leads, in one way or the other to dimensionality +reductions. Most of the Machine Learning methods we encounter deal +with this, whether we opt for a principal component analysis, or +clustering, or convolutional neural networks, or Ridge or Lasso +regression or random forest, yes, perhaps most machine learning +methods at large. + +

    +For neural networks and our previous discussion, we have seen that we +in essence end up with matrix-matrix and matrix-vector +multiplications. In all cases, our matrices are dense ones, and the +more data we deal with the larger the dimensionalities of the matrices +and vectors. How can we reduce such dimensionalities? One possible +answer is offered by convolutional neural networks (CNN), as +discussed below. The figure here shows a typical situation of the +reduction of information in an image and is typical of what CNNs +actually end up doing. + +

    +









    + +

    From a Spherical Cow to a real one

    + +

    +



    + +

    +









    +

    Convolutional Neural Networks (recognizing images)

    diff --git a/doc/pub/week41/html/week41.html b/doc/pub/week41/html/week41.html index d676fed29..c152409d4 100644 --- a/doc/pub/week41/html/week41.html +++ b/doc/pub/week41/html/week41.html @@ -132,6 +132,19 @@ div { text-align: justify; text-justify: inter-word; } None, 'scikit-learn-implementation'), ('Visualization', 2, None, 'visualization'), + ('Testing our code for the XOR, OR and AND gates', + 2, + None, + 'testing-our-code-for-the-xor-or-and-and-gates'), + ('The AND and XOR Gates', 2, None, 'the-and-and-xor-gates'), + ('Representing the Data Sets', + 2, + None, + 'representing-the-data-sets'), + ('Setting up the Neural Network', + 2, + None, + 'setting-up-the-neural-network'), ('Building neural networks in Tensorflow and Keras', 2, None, @@ -187,6 +200,14 @@ div { text-align: justify; text-justify: inter-word; } 2, None, 'limitations-of-supervised-learning-with-deep-networks'), + ('Overarching Views, a personal note', + 2, + None, + 'overarching-views-a-personal-note'), + ('From a Spherical Cow to a real one', + 2, + None, + 'from-a-spherical-cow-to-a-real-one'), ('Convolutional Neural Networks (recognizing images)', 2, None, @@ -265,7 +286,7 @@ MathJax.Hub.Config({

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

    -

    Oct 11, 2021

    +

    Oct 13, 2021












    @@ -1462,6 +1483,110 @@ plt.show()











    +

    Testing our code for the XOR, OR and AND gates

    + +

    +Last week we discussed three different types of gates, the so-called +XOR, the OR and the AND gates. Their inputs and outputs can be +summarized using the following tables, first for the OR gate with +inputs \( x_1 \) and \( x_2 \) and outputs \( y \): + +

    + + + + + + + + + + +
    \( x_1 \) \( x_2 \) \( y \)
    0 0 0
    0 1 1
    1 0 1
    1 1 1
    +

    +









    + +

    The AND and XOR Gates

    + +

    +The AND gate is defined as + +

    + + + + + + + + + + +
    \( x_1 \) \( x_2 \) \( y \)
    0 0 0
    0 1 0
    1 0 0
    1 1 1
    +

    +And finally we have the XOR gate + +

    + + + + + + + + + + +
    \( x_1 \) \( x_2 \) \( y \)
    0 0 0
    0 1 1
    1 0 1
    1 1 0
    +

    +









    + +

    Representing the Data Sets

    + +

    +Our design matrix is defined by the input values \( x_1 \) and \( x_2 \). Since we have four possible outputs, our design matrix reads + +$$ +\boldsymbol{X}=\begin{bmatrix} 0 & 0 \\ + 0 & 1 \\ + 1 & 0 \\ + 1 & 1 \end{bmatrix}, +$$ + +while the vector of outputs is \( \boldsymbol{y}^T=[0,1,1,0] \) for the XOR gate, \( \boldsymbol{y}^T=[0,0,0,1] \) for the AND gate and \( \boldsymbol{y}^T=[0,1,1,1] \) for the OR gate. + +

    +









    + +

    Setting up the Neural Network

    + +

    +We define first our design matrix and the various input vectors. + +

    + + +

    """
    +Simple code that tests XOR, OR and AND gates with linear regression
    +"""
    +
    +import numpy as np
    +# Design matrix
    +X = np.array([ [1, 0, 0], [1, 0, 1], [1, 1, 0],[1, 1, 1]],dtype=np.float64)
    +
    +# The XOR gate 
    +yXOR = np.array( [ 0, 1 ,1, 0])
    +# The OR gate 
    +yOR = np.array( [ 0, 1 ,1, 1])
    +# The AND gate 
    +yAND = np.array( [ 0, 0 ,0, 1])
    +
    +#print(f"The values of theta for the AND gate:{ThetaAND}")
    +#print(f"The linear regression prediction  for the AND gate:{X @ ThetaAND}")
    +
    +

    +









    +

    Building neural networks in Tensorflow and Keras

    @@ -2210,6 +2335,45 @@ Some of these remarks are particular to DNNs, others are shared by all supervise











    +

    Overarching Views, a personal note

    + +

    +The author of these lecture notes has an overarching take on many of +the machine learning algorithms we discuss here. + +

    +If we wish to understand complex systems, we need to find some +effective degrees of freedom or features that we find essential, +simply in order to reduce the complexity of the systems we are +studying. This leads, in one way or the other to dimensionality +reductions. Most of the Machine Learning methods we encounter deal +with this, whether we opt for a principal component analysis, or +clustering, or convolutional neural networks, or Ridge or Lasso +regression or random forest, yes, perhaps most machine learning +methods at large. + +

    +For neural networks and our previous discussion, we have seen that we +in essence end up with matrix-matrix and matrix-vector +multiplications. In all cases, our matrices are dense ones, and the +more data we deal with the larger the dimensionalities of the matrices +and vectors. How can we reduce such dimensionalities? One possible +answer is offered by convolutional neural networks (CNN), as +discussed below. The figure here shows a typical situation of the +reduction of information in an image and is typical of what CNNs +actually end up doing. + +

    +









    + +

    From a Spherical Cow to a real one

    + +

    +



    + +

    +









    +

    Convolutional Neural Networks (recognizing images)

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"**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 11, 2021**\n", + "Date: **Oct 13, 2021**\n", "\n", "Copyright 1999-2021, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license\n", "\n", @@ -1383,6 +1383,112 @@ "plt.show()" ] }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Testing our code for the XOR, OR and AND gates\n", + "\n", + "Last week we discussed three different types of gates, the so-called\n", + "XOR, the OR and the AND gates. Their inputs and outputs can be\n", + "summarized using the following tables, first for the OR gate with\n", + "inputs $x_1$ and $x_2$ and outputs $y$:\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "
        $x_1$ $x_2$ $y$
        0 0 0
        0 1 1
        1 0 1
        1 1 1
        \n", + "## The AND and XOR Gates\n", + "\n", + "The **AND** gate is defined as\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "
        $x_1$ $x_2$ $y$
        0 0 0
        0 1 0
        1 0 0
        1 1 1
        \n", + "And finally we have the **XOR** gate\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "
        $x_1$ $x_2$ $y$
        0 0 0
        0 1 1
        1 0 1
        1 1 0
        \n", + "## Representing the Data Sets\n", + "\n", + "Our design matrix is defined by the input values $x_1$ and $x_2$. Since we have four possible outputs, our design matrix reads" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\boldsymbol{X}=\\begin{bmatrix} 0 & 0 \\\\\n", + " 0 & 1 \\\\\n", + "\t\t 1 & 0 \\\\\n", + "\t\t 1 & 1 \\end{bmatrix},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "while the vector of outputs is $\\boldsymbol{y}^T=[0,1,1,0]$ for the XOR gate, $\\boldsymbol{y}^T=[0,0,0,1]$ for the AND gate and $\\boldsymbol{y}^T=[0,1,1,1]$ for the OR gate.\n", + "\n", + "## Setting up the Neural Network\n", + "\n", + "We define first our design matrix and the various input vectors." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "\"\"\"\n", + "Simple code that tests XOR, OR and AND gates with linear regression\n", + "\"\"\"\n", + "\n", + "import numpy as np\n", + "# Design matrix\n", + "X = np.array([ [1, 0, 0], [1, 0, 1], [1, 1, 0],[1, 1, 1]],dtype=np.float64)\n", + "\n", + "# The XOR gate \n", + "yXOR = np.array( [ 0, 1 ,1, 0])\n", + "# The OR gate \n", + "yOR = np.array( [ 0, 1 ,1, 1])\n", + "# The AND gate \n", + "yAND = np.array( [ 0, 0 ,0, 1])\n", + "\n", + "#print(f\"The values of theta for the AND gate:{ThetaAND}\")\n", + "#print(f\"The linear regression prediction for the AND gate:{X @ ThetaAND}\")" + ] + }, { "cell_type": "markdown", "metadata": {}, @@ -2171,6 +2277,37 @@ "\n", "\n", "\n", + "## Overarching Views, a personal note\n", + "\n", + "The author of these lecture notes has an overarching take on many of\n", + "the machine learning algorithms we discuss here. \n", + "\n", + "\n", + "If we wish to understand complex systems, we need to find some\n", + "effective degrees of freedom or features that we find essential,\n", + "simply in order to reduce the complexity of the systems we are\n", + "studying. This leads, in one way or the other to dimensionality\n", + "reductions. Most of the Machine Learning methods we encounter deal\n", + "with this, whether we opt for a principal component analysis, or\n", + "clustering, or convolutional neural networks, or Ridge or Lasso\n", + "regression or random forest, yes, perhaps most machine learning\n", + "methods at large.\n", + "\n", + "For neural networks and our previous discussion, we have seen that we\n", + "in essence end up with matrix-matrix and matrix-vector\n", + "multiplications. In all cases, our matrices are dense ones, and the\n", + "more data we deal with the larger the dimensionalities of the matrices\n", + "and vectors. How can we reduce such dimensionalities? One possible\n", + "answer is offered by **convolutional neural networks** (CNN), as\n", + "discussed below. The figure here shows a typical situation of the\n", + "reduction of information in an image and is typical of what CNNs\n", + "actually end up doing.\n", + "\n", + "## From a Spherical Cow to a real one\n", + "\n", + "\n", + "\n", + "

        Figure 1:

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for the XOR, OR and AND gates ===== + +Last week we discussed three different types of gates, the so-called +XOR, the OR and the AND gates. Their inputs and outputs can be +summarized using the following tables, first for the OR gate with +inputs $x_1$ and $x_2$ and outputs $y$: + +|---------------------| +| $x_1$ | $x_2$ | $y$ | +|---------------------| +| 0 | 0 | 0 | +| 0 | 1 | 1 | +| 1 | 0 | 1 | +| 1 | 1 | 1 | +|---------------------| + +!split +===== The AND and XOR Gates ===== + +The _AND_ gate is defined as + +|---------------------| +| $x_1$ | $x_2$ | $y$ | +|---------------------| +| 0 | 0 | 0 | +| 0 | 1 | 0 | +| 1 | 0 | 0 | +| 1 | 1 | 1 | +|---------------------| + +And finally we have the _XOR_ gate + +|---------------------| +| $x_1$ | $x_2$ | $y$ | +|---------------------| +| 0 | 0 | 0 | +| 0 | 1 | 1 | +| 1 | 0 | 1 | +| 1 | 1 | 0 | +|---------------------| + +!split +===== Representing the Data Sets ===== + +Our design matrix is defined by the input values $x_1$ and $x_2$. Since we have four possible outputs, our design matrix reads + +!bt +\bm{X}=\begin{bmatrix} 0 & 0 \\ + 0 & 1 \\ + 1 & 0 \\ + 1 & 1 \end{bmatrix}, +!et +while the vector of outputs is $\bm{y}^T=[0,1,1,0]$ for the XOR gate, $\bm{y}^T=[0,0,0,1]$ for the AND gate and $\bm{y}^T=[0,1,1,1]$ for the OR gate. + +!split +===== Setting up the Neural Network ===== + +We define first our design matrix and the various input vectors. + +!bc pycod +""" +Simple code that tests XOR, OR and AND gates with linear regression +""" + +import numpy as np +# Design matrix +X = np.array([ [1, 0, 0], [1, 0, 1], [1, 1, 0],[1, 1, 1]],dtype=np.float64) + +# The XOR gate +yXOR = np.array( [ 0, 1 ,1, 0]) +# The OR gate +yOR = np.array( [ 0, 1 ,1, 1]) +# The AND gate +yAND = np.array( [ 0, 0 ,0, 1]) + +#print(f"The values of theta for the AND gate:{ThetaAND}") +#print(f"The linear regression prediction for the AND gate:{X @ ThetaAND}") +!ec + + + !split ===== Building neural networks in Tensorflow and Keras ===== @@ -1749,7 +1831,37 @@ Some of these remarks are particular to DNNs, others are shared by all supervise +!split +===== Overarching Views, a personal note ===== +The author of these lecture notes has an overarching take on many of +the machine learning algorithms we discuss here. + + +If we wish to understand complex systems, we need to find some +effective degrees of freedom or features that we find essential, +simply in order to reduce the complexity of the systems we are +studying. This leads, in one way or the other to dimensionality +reductions. Most of the Machine Learning methods we encounter deal +with this, whether we opt for a principal component analysis, or +clustering, or convolutional neural networks, or Ridge or Lasso +regression or random forest, yes, perhaps most machine learning +methods at large. + +For neural networks and our previous discussion, we have seen that we +in essence end up with matrix-matrix and matrix-vector +multiplications. In all cases, our matrices are dense ones, and the +more data we deal with the larger the dimensionalities of the matrices +and vectors. How can we reduce such dimensionalities? One possible +answer is offered by _convolutional neural networks_ (CNN), as +discussed below. The figure here shows a typical situation of the +reduction of information in an image and is typical of what CNNs +actually end up doing. + +!split +===== From a Spherical Cow to a real one ===== + +FIGURE: [figslides/ImageReduction.png, width=500 frac=0.6] !split @@ -2148,3 +2260,5 @@ o "Abstract art using convolutional neural networks":"https://deepdreamgenerator + +

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