From 06d468b2d4caefeb0d189b1a2d99bd8ca7a60844 Mon Sep 17 00:00:00 2001 From: mhjensen Date: Wed, 22 Nov 2017 15:51:08 +0100 Subject: [PATCH] Update on NN --- doc/pub/NeuralNet/html/NeuralNet-bs.html | 153 +++++++++++++++--- doc/pub/NeuralNet/html/NeuralNet-reveal.html | 151 +++++++++++++---- .../NeuralNet/html/NeuralNet-solarized.html | 141 +++++++++++++--- doc/pub/NeuralNet/html/NeuralNet.html | 141 +++++++++++++--- doc/pub/NeuralNet/ipynb/NeuralNet.ipynb | 138 ++++++++++++---- .../ipynb/ipynb-NeuralNet-src.tar.gz | Bin 211 -> 212 bytes .../pdf/NeuralNet-beamer-handouts2x3.pdf | Bin 185707 -> 196526 bytes doc/pub/NeuralNet/pdf/NeuralNet-beamer.pdf | Bin 184164 -> 192481 bytes doc/pub/NeuralNet/pdf/NeuralNet-minted.pdf | Bin 252799 -> 259895 bytes doc/src/NeuralNet/NeuralNet.do.txt | 99 ++++++++++-- 10 files changed, 679 insertions(+), 144 deletions(-) diff --git a/doc/pub/NeuralNet/html/NeuralNet-bs.html b/doc/pub/NeuralNet/html/NeuralNet-bs.html index c286d1928..4f1a9251e 100644 --- a/doc/pub/NeuralNet/html/NeuralNet-bs.html +++ b/doc/pub/NeuralNet/html/NeuralNet-bs.html @@ -47,12 +47,21 @@ Automatically generated HTML file from DocOnce source ('Feed-forward neural networks', 2, None, '___sec4'), ('Recurrent neural networks', 2, None, '___sec5'), ('Other types of networks', 2, None, '___sec6'), - ('Mathematical model', 2, None, '___sec7'), - ('Mathematical model', 2, None, '___sec8'), + ('Multilayer perceptrons', 2, None, '___sec7'), + ('Why multilayer perceptrons?', 2, None, '___sec8'), ('Mathematical model', 2, None, '___sec9'), ('Mathematical model', 2, None, '___sec10'), ('Mathematical model', 2, None, '___sec11'), - ('Matrix-vector notation', 3, None, '___sec12')]} + ('Mathematical model', 2, None, '___sec12'), + ('Mathematical model', 2, None, '___sec13'), + ('Matrix-vector notation', 3, None, '___sec14'), + ('Matrix-vector notation and activation', 3, None, '___sec15'), + ('Activation functions', 3, None, '___sec16'), + ('Activation functions, Logistic and Hyperbolic ones', + 3, + None, + '___sec17'), + ('Relevance', 3, None, '___sec18')]} end of tocinfo --> @@ -97,12 +106,18 @@ MathJax.Hub.Config({
  • Feed-forward neural networks
  • Recurrent neural networks
  • Other types of networks
  • -
  • Mathematical model
  • -
  • Mathematical model
  • +
  • Multilayer perceptrons
  • +
  • Why multilayer perceptrons?
  • Mathematical model
  • Mathematical model
  • Mathematical model
  • -
  •    Matrix-vector notation
  • +
  • Mathematical model
  • +
  • Mathematical model
  • +
  •    Matrix-vector notation
  • +
  •    Matrix-vector notation and activation
  • +
  •    Activation functions
  • +
  •    Activation functions, Logistic and Hyperbolic ones
  • +
  •    Relevance
  • @@ -283,7 +298,41 @@ of how a fully-connected FFNN works, and how it can be used to interpolate data

    -

    Mathematical model

    +

    Multilayer perceptrons

    + +

    +One use often so-called fully-connected feed-forward neural networks with three +or more layers (an input layer, one or more hidden layers and an output layer) +consisting of neurons that have non-linear activation functions. + +

    +Such networks are often called multilayer perceptrons (MLPs) + +

    + + +

    Why multilayer perceptrons?

    + +

    +According to the Universal approximation theorem, a feed-forward neural network with just a single hidden layer containing +a finite number of neurons can approximate a continuous multidimensional function to arbitrary accuracy, +assuming the activation function for the hidden layer is a non-constant, bounded and monotonically-increasing continuous function. +Note that the requirements on the activation function only applies to the hidden layer, the output nodes are always +assumed to be linear, so as to not restrict the range of output values. + +

    +We note that this theorem is only applicable to a NN with one hidden layer. +Therefore, we can easily construct an NN +that employs activation functions which do not satisfy the above requirements, as long as we have at least one layer +with activation functions that do. Furthermore, although the universal approximation theorem +lays the theoretical foundation for regression with neural networks, it does not say anything about how things work in practice: +A neural network can still be able to approximate a given function reasonably well without having the flexibility to fit all other +functions. + +

    + + +

    Mathematical model

    $$ \begin{equation} @@ -293,13 +342,13 @@ $$ $$ In an FFNN of such neurons, the inputs \( x_i \) -are the outputs of the neurons in the preceding layer. Furthermore, a MLP is fully-connected, +are the outputs of the neurons in the preceding layer. Furthermore, an MLP is fully-connected, which means that each neuron receives a weighted sum of the outputs of all neurons in the previous layer.

    -

    Mathematical model

    +

    Mathematical model

    First, for each node \( i \) in the first hidden layer, we calculate a weighted sum \( u_i^1 \) of the input coordinates \( x_j \), @@ -336,7 +385,7 @@ the values of the subsequent layer can be calculated and so forth until the outp

    -

    Mathematical model

    +

    Mathematical model

    The output of neuron \( i \) in layer 2 is thus, @@ -365,7 +414,7 @@ $$

    -

    Mathematical model

    +

    Mathematical model

    We can generalize this expression to an MLP with \( l \) hidden layers. The complete functional form @@ -385,7 +434,7 @@ which illustrates a basic property of MLPs: The only independent variables are t

    -

    Mathematical model

    +

    Mathematical model

    This confirms that an MLP, @@ -407,19 +456,12 @@ $$ where the parameters \( c_i \) are weights and biases. By adjusting these parameters, the activation functions can be shifted up and down or left and right, change slope or be rescaled -which is the key to the flexibility of a NN. - -$$ -\begin{equation} - f_o = f(u_o) = u_o -\label{outputActivation} -\end{equation} -$$ +which is the key to the flexibility of a neural network.

    -

    Matrix-vector notation

    +

    Matrix-vector notation

    We can introduce a more convenient notation for the activations in a NN. @@ -432,7 +474,6 @@ is the bias \( b_i^l \) and activation \( y_i^l \) of node \( i \) in layer \( l We have that \( \mathrm{W}_l \) is a \( N_{l-1} \times N_l \) matrix, while \( \vec{b}_l \) and \( \vec{y}_l \) are \( N_l \times 1 \) column vectors. With this notation, the sum in becomes a matrix-vector multiplication, and we can write the equation for the activations of hidden layer 2 in - $$ \begin{equation} \vec{y}_2 = f_2(\mathrm{W}_2 \vec{y}_{1} + \vec{b}_{2}) = @@ -450,27 +491,87 @@ $$ b^2_1 \\ b^2_2 \\ b^2_3 \\ - \end{array}\right]\right) + \end{array}\right]\right). \label{_auto6} \end{equation} $$ -and we see that the activation of node \( i \) in layer 2 is +

    + + +

    Matrix-vector notation and activation

    + +The activation of node \( i \) in layer 2 is $$ \begin{equation} y^2_i = f_2\Bigr(w^2_{i1}y^1_1 + w^2_{i2}y^1_2 + w^2_{i3}y^1_3 + b^2_i\Bigr) = - f_2\left(\sum_{j=1}^3 w^2_{ij} y_j^1 + b^2_i\right) + f_2\left(\sum_{j=1}^3 w^2_{ij} y_j^1 + b^2_i\right). \label{_auto7} \end{equation} $$ -which is in accordance with. Note that This is not just a convenient and compact notation, but also a useful and intuitive way to think about MLPs: The output is calculated by a series of matrix-vector multiplications and vector additions that are used as input to the activation functions. For each operation \( \mathrm{W}_l \vec{y}_{l-1} \) we move forward one layer. +

    + + +

    Activation functions

    + +

    +A property that characterizes a neural network, other than its connectivity, is the choice of activation function(s). +As described in, the following restrictions are imposed on an activation function for a FFNN +to fulfill the universal approximation theorem + +

    + + + +

    Activation functions, Logistic and Hyperbolic ones

    + +

    +The second requirement excludes all linear functions. Furthermore, in a MLP with only linear activation functions, each +layer simply performs a linear transformation of its inputs. + +

    +Regardless of the number of layers, +the output of the NN will be nothing but a linear function of the inputs. Thus we need to introduce some kind of +non-linearity to the NN to be able to fit non-linear functions +Typical examples are the logistic Sigmoid +$$ +\begin{equation} + f(x) = \frac{1}{1 + e^{-x}}, +\label{sigmoidActivationFunction} +\end{equation} +$$ + +and the hyperbolic tangent function +$$ +\begin{equation} + f(x) = \tanh(x) +\label{tanhActivationFunction} +\end{equation} +$$ + +

    + + +

    Relevance

    + +The sigmoid function are more biologically plausible because +the output of inactive neurons are zero. Such activation function are called one-sided. However, +it has been shown that the hyperbolic tangent +performs better than the sigmoid for training MLPs. +has become the most popular for deep neural networks +

    diff --git a/doc/pub/NeuralNet/html/NeuralNet-reveal.html b/doc/pub/NeuralNet/html/NeuralNet-reveal.html index 6e4dcb45b..84cc3cc01 100644 --- a/doc/pub/NeuralNet/html/NeuralNet-reveal.html +++ b/doc/pub/NeuralNet/html/NeuralNet-reveal.html @@ -304,7 +304,41 @@ of how a fully-connected FFNN works, and how it can be used to interpolate data

    -

    Mathematical model

    +

    Multilayer perceptrons

    + +

    +One use often so-called fully-connected feed-forward neural networks with three +or more layers (an input layer, one or more hidden layers and an output layer) +consisting of neurons that have non-linear activation functions. + +

    +Such networks are often called multilayer perceptrons (MLPs) +

    + + +
    +

    Why multilayer perceptrons?

    + +

    +According to the Universal approximation theorem, a feed-forward neural network with just a single hidden layer containing +a finite number of neurons can approximate a continuous multidimensional function to arbitrary accuracy, +assuming the activation function for the hidden layer is a non-constant, bounded and monotonically-increasing continuous function. +Note that the requirements on the activation function only applies to the hidden layer, the output nodes are always +assumed to be linear, so as to not restrict the range of output values. + +

    +We note that this theorem is only applicable to a NN with one hidden layer. +Therefore, we can easily construct an NN +that employs activation functions which do not satisfy the above requirements, as long as we have at least one layer +with activation functions that do. Furthermore, although the universal approximation theorem +lays the theoretical foundation for regression with neural networks, it does not say anything about how things work in practice: +A neural network can still be able to approximate a given function reasonably well without having the flexibility to fit all other +functions. +

    + + +
    +

    Mathematical model

     
    $$ @@ -316,13 +350,13 @@ $$

     
    In an FFNN of such neurons, the inputs \( x_i \) -are the outputs of the neurons in the preceding layer. Furthermore, a MLP is fully-connected, +are the outputs of the neurons in the preceding layer. Furthermore, an MLP is fully-connected, which means that each neuron receives a weighted sum of the outputs of all neurons in the previous layer.

    -

    Mathematical model

    +

    Mathematical model

    First, for each node \( i \) in the first hidden layer, we calculate a weighted sum \( u_i^1 \) of the input coordinates \( x_j \), @@ -365,7 +399,7 @@ the values of the subsequent layer can be calculated and so forth until the outp

    -

    Mathematical model

    +

    Mathematical model

    The output of neuron \( i \) in layer 2 is thus, @@ -398,7 +432,7 @@ $$

    -

    Mathematical model

    +

    Mathematical model

    We can generalize this expression to an MLP with \( l \) hidden layers. The complete functional form @@ -420,7 +454,7 @@ which illustrates a basic property of MLPs: The only independent variables are t

    -

    Mathematical model

    +

    Mathematical model

    This confirms that an MLP, @@ -444,22 +478,13 @@ $$ where the parameters \( c_i \) are weights and biases. By adjusting these parameters, the activation functions can be shifted up and down or left and right, change slope or be rescaled -which is the key to the flexibility of a NN. - -

     
    -$$ -\begin{equation} - f_o = f(u_o) = u_o -\tag{12} -\end{equation} -$$ -

     
    +which is the key to the flexibility of a neural network.

    -

    Matrix-vector notation

    +

    Matrix-vector notation

    We can introduce a more convenient notation for the activations in a NN. @@ -472,7 +497,6 @@ is the bias \( b_i^l \) and activation \( y_i^l \) of node \( i \) in layer \( l We have that \( \mathrm{W}_l \) is a \( N_{l-1} \times N_l \) matrix, while \( \vec{b}_l \) and \( \vec{y}_l \) are \( N_l \times 1 \) column vectors. With this notation, the sum in becomes a matrix-vector multiplication, and we can write the equation for the activations of hidden layer 2 in -

     
    $$ \begin{equation} @@ -491,29 +515,102 @@ $$ b^2_1 \\ b^2_2 \\ b^2_3 \\ - \end{array}\right]\right) -\tag{13} + \end{array}\right]\right). +\tag{12} \end{equation} $$

     
    +

    -and we see that the activation of node \( i \) in layer 2 is + +
    + +

    Matrix-vector notation and activation

    + +The activation of node \( i \) in layer 2 is

     
    $$ \begin{equation} y^2_i = f_2\Bigr(w^2_{i1}y^1_1 + w^2_{i2}y^1_2 + w^2_{i3}y^1_3 + b^2_i\Bigr) = - f_2\left(\sum_{j=1}^3 w^2_{ij} y_j^1 + b^2_i\right) + f_2\left(\sum_{j=1}^3 w^2_{ij} y_j^1 + b^2_i\right). +\tag{13} +\end{equation} +$$ +

     
    + +This is not just a convenient and compact notation, but also +a useful and intuitive way to think about MLPs: The output is calculated by a series of matrix-vector multiplications +and vector additions that are used as input to the activation functions. For each operation +\( \mathrm{W}_l \vec{y}_{l-1} \) we move forward one layer. +

    + + +
    + +

    Activation functions

    + +

    +A property that characterizes a neural network, other than its connectivity, is the choice of activation function(s). +As described in, the following restrictions are imposed on an activation function for a FFNN +to fulfill the universal approximation theorem + +

      + +

    • Non-constant
    • + +

    • Bounded
    • + +

    • Monotonically-increasing
    • + +

    • Continuous
    • +
    +
    + + +
    + +

    Activation functions, Logistic and Hyperbolic ones

    + +

    +The second requirement excludes all linear functions. Furthermore, in a MLP with only linear activation functions, each +layer simply performs a linear transformation of its inputs. + +

    +Regardless of the number of layers, +the output of the NN will be nothing but a linear function of the inputs. Thus we need to introduce some kind of +non-linearity to the NN to be able to fit non-linear functions +Typical examples are the logistic Sigmoid +

     
    +$$ +\begin{equation} + f(x) = \frac{1}{1 + e^{-x}}, \tag{14} \end{equation} $$

     
    -which is in accordance with. Note that -This is not just a convenient and compact notation, but also -a useful and intuitive way to think about MLPs: The output is calculated by a series of matrix-vector multiplications -and vector additions that are used as input to the activation functions. For each operation -\( \mathrm{W}_l \vec{y}_{l-1} \) we move forward one layer. +and the hyperbolic tangent function +

     
    +$$ +\begin{equation} + f(x) = \tanh(x) +\tag{15} +\end{equation} +$$ +

     
    +

    + + +
    + +

    Relevance

    + +The sigmoid function are more biologically plausible because +the output of inactive neurons are zero. Such activation function are called one-sided. However, +it has been shown that the hyperbolic tangent +performs better than the sigmoid for training MLPs. +has become the most popular for deep neural networks
    diff --git a/doc/pub/NeuralNet/html/NeuralNet-solarized.html b/doc/pub/NeuralNet/html/NeuralNet-solarized.html index fa48edec1..3f3ebf526 100644 --- a/doc/pub/NeuralNet/html/NeuralNet-solarized.html +++ b/doc/pub/NeuralNet/html/NeuralNet-solarized.html @@ -41,12 +41,21 @@ div { text-align: justify; text-justify: inter-word; } ('Feed-forward neural networks', 2, None, '___sec4'), ('Recurrent neural networks', 2, None, '___sec5'), ('Other types of networks', 2, None, '___sec6'), - ('Mathematical model', 2, None, '___sec7'), - ('Mathematical model', 2, None, '___sec8'), + ('Multilayer perceptrons', 2, None, '___sec7'), + ('Why multilayer perceptrons?', 2, None, '___sec8'), ('Mathematical model', 2, None, '___sec9'), ('Mathematical model', 2, None, '___sec10'), ('Mathematical model', 2, None, '___sec11'), - ('Matrix-vector notation', 3, None, '___sec12')]} + ('Mathematical model', 2, None, '___sec12'), + ('Mathematical model', 2, None, '___sec13'), + ('Matrix-vector notation', 3, None, '___sec14'), + ('Matrix-vector notation and activation', 3, None, '___sec15'), + ('Activation functions', 3, None, '___sec16'), + ('Activation functions, Logistic and Hyperbolic ones', + 3, + None, + '___sec17'), + ('Relevance', 3, None, '___sec18')]} end of tocinfo --> @@ -232,7 +241,41 @@ of how a fully-connected FFNN works, and how it can be used to interpolate data











    -

    Mathematical model

    +

    Multilayer perceptrons

    + +

    +One use often so-called fully-connected feed-forward neural networks with three +or more layers (an input layer, one or more hidden layers and an output layer) +consisting of neurons that have non-linear activation functions. + +

    +Such networks are often called multilayer perceptrons (MLPs) + +

    +









    + +

    Why multilayer perceptrons?

    + +

    +According to the Universal approximation theorem, a feed-forward neural network with just a single hidden layer containing +a finite number of neurons can approximate a continuous multidimensional function to arbitrary accuracy, +assuming the activation function for the hidden layer is a non-constant, bounded and monotonically-increasing continuous function. +Note that the requirements on the activation function only applies to the hidden layer, the output nodes are always +assumed to be linear, so as to not restrict the range of output values. + +

    +We note that this theorem is only applicable to a NN with one hidden layer. +Therefore, we can easily construct an NN +that employs activation functions which do not satisfy the above requirements, as long as we have at least one layer +with activation functions that do. Furthermore, although the universal approximation theorem +lays the theoretical foundation for regression with neural networks, it does not say anything about how things work in practice: +A neural network can still be able to approximate a given function reasonably well without having the flexibility to fit all other +functions. + +

    +









    + +

    Mathematical model

    $$ \begin{equation} @@ -242,13 +285,13 @@ $$ $$ In an FFNN of such neurons, the inputs \( x_i \) -are the outputs of the neurons in the preceding layer. Furthermore, a MLP is fully-connected, +are the outputs of the neurons in the preceding layer. Furthermore, an MLP is fully-connected, which means that each neuron receives a weighted sum of the outputs of all neurons in the previous layer.











    -

    Mathematical model

    +

    Mathematical model

    First, for each node \( i \) in the first hidden layer, we calculate a weighted sum \( u_i^1 \) of the input coordinates \( x_j \), @@ -285,7 +328,7 @@ the values of the subsequent layer can be calculated and so forth until the outp











    -

    Mathematical model

    +

    Mathematical model

    The output of neuron \( i \) in layer 2 is thus, @@ -314,7 +357,7 @@ $$











    -

    Mathematical model

    +

    Mathematical model

    We can generalize this expression to an MLP with \( l \) hidden layers. The complete functional form @@ -334,7 +377,7 @@ which illustrates a basic property of MLPs: The only independent variables are t











    -

    Mathematical model

    +

    Mathematical model

    This confirms that an MLP, @@ -356,19 +399,12 @@ $$ where the parameters \( c_i \) are weights and biases. By adjusting these parameters, the activation functions can be shifted up and down or left and right, change slope or be rescaled -which is the key to the flexibility of a NN. - -$$ -\begin{equation} - f_o = f(u_o) = u_o -\label{outputActivation} -\end{equation} -$$ +which is the key to the flexibility of a neural network.











    -

    Matrix-vector notation

    +

    Matrix-vector notation

    We can introduce a more convenient notation for the activations in a NN. @@ -381,7 +417,6 @@ is the bias \( b_i^l \) and activation \( y_i^l \) of node \( i \) in layer \( l We have that \( \mathrm{W}_l \) is a \( N_{l-1} \times N_l \) matrix, while \( \vec{b}_l \) and \( \vec{y}_l \) are \( N_l \times 1 \) column vectors. With this notation, the sum in becomes a matrix-vector multiplication, and we can write the equation for the activations of hidden layer 2 in - $$ \begin{equation} \vec{y}_2 = f_2(\mathrm{W}_2 \vec{y}_{1} + \vec{b}_{2}) = @@ -399,27 +434,87 @@ $$ b^2_1 \\ b^2_2 \\ b^2_3 \\ - \end{array}\right]\right) + \end{array}\right]\right). \label{_auto6} \end{equation} $$ -and we see that the activation of node \( i \) in layer 2 is +

    +









    + +

    Matrix-vector notation and activation

    + +The activation of node \( i \) in layer 2 is $$ \begin{equation} y^2_i = f_2\Bigr(w^2_{i1}y^1_1 + w^2_{i2}y^1_2 + w^2_{i3}y^1_3 + b^2_i\Bigr) = - f_2\left(\sum_{j=1}^3 w^2_{ij} y_j^1 + b^2_i\right) + f_2\left(\sum_{j=1}^3 w^2_{ij} y_j^1 + b^2_i\right). \label{_auto7} \end{equation} $$ -which is in accordance with. Note that This is not just a convenient and compact notation, but also a useful and intuitive way to think about MLPs: The output is calculated by a series of matrix-vector multiplications and vector additions that are used as input to the activation functions. For each operation \( \mathrm{W}_l \vec{y}_{l-1} \) we move forward one layer. +

    +









    + +

    Activation functions

    + +

    +A property that characterizes a neural network, other than its connectivity, is the choice of activation function(s). +As described in, the following restrictions are imposed on an activation function for a FFNN +to fulfill the universal approximation theorem + +

      +
    • Non-constant
    • +
    • Bounded
    • +
    • Monotonically-increasing
    • +
    • Continuous
    • +
    + +









    + +

    Activation functions, Logistic and Hyperbolic ones

    + +

    +The second requirement excludes all linear functions. Furthermore, in a MLP with only linear activation functions, each +layer simply performs a linear transformation of its inputs. + +

    +Regardless of the number of layers, +the output of the NN will be nothing but a linear function of the inputs. Thus we need to introduce some kind of +non-linearity to the NN to be able to fit non-linear functions +Typical examples are the logistic Sigmoid +$$ +\begin{equation} + f(x) = \frac{1}{1 + e^{-x}}, +\label{sigmoidActivationFunction} +\end{equation} +$$ + +and the hyperbolic tangent function +$$ +\begin{equation} + f(x) = \tanh(x) +\label{tanhActivationFunction} +\end{equation} +$$ + +

    +









    + +

    Relevance

    + +The sigmoid function are more biologically plausible because +the output of inactive neurons are zero. Such activation function are called one-sided. However, +it has been shown that the hyperbolic tangent +performs better than the sigmoid for training MLPs. +has become the most popular for deep neural networks +

    diff --git a/doc/pub/NeuralNet/html/NeuralNet.html b/doc/pub/NeuralNet/html/NeuralNet.html index 04804c9c2..c43db64cc 100644 --- a/doc/pub/NeuralNet/html/NeuralNet.html +++ b/doc/pub/NeuralNet/html/NeuralNet.html @@ -46,12 +46,21 @@ div { text-align: justify; text-justify: inter-word; } ('Feed-forward neural networks', 2, None, '___sec4'), ('Recurrent neural networks', 2, None, '___sec5'), ('Other types of networks', 2, None, '___sec6'), - ('Mathematical model', 2, None, '___sec7'), - ('Mathematical model', 2, None, '___sec8'), + ('Multilayer perceptrons', 2, None, '___sec7'), + ('Why multilayer perceptrons?', 2, None, '___sec8'), ('Mathematical model', 2, None, '___sec9'), ('Mathematical model', 2, None, '___sec10'), ('Mathematical model', 2, None, '___sec11'), - ('Matrix-vector notation', 3, None, '___sec12')]} + ('Mathematical model', 2, None, '___sec12'), + ('Mathematical model', 2, None, '___sec13'), + ('Matrix-vector notation', 3, None, '___sec14'), + ('Matrix-vector notation and activation', 3, None, '___sec15'), + ('Activation functions', 3, None, '___sec16'), + ('Activation functions, Logistic and Hyperbolic ones', + 3, + None, + '___sec17'), + ('Relevance', 3, None, '___sec18')]} end of tocinfo --> @@ -237,7 +246,41 @@ of how a fully-connected FFNN works, and how it can be used to interpolate data











    -

    Mathematical model

    +

    Multilayer perceptrons

    + +

    +One use often so-called fully-connected feed-forward neural networks with three +or more layers (an input layer, one or more hidden layers and an output layer) +consisting of neurons that have non-linear activation functions. + +

    +Such networks are often called multilayer perceptrons (MLPs) + +

    +









    + +

    Why multilayer perceptrons?

    + +

    +According to the Universal approximation theorem, a feed-forward neural network with just a single hidden layer containing +a finite number of neurons can approximate a continuous multidimensional function to arbitrary accuracy, +assuming the activation function for the hidden layer is a non-constant, bounded and monotonically-increasing continuous function. +Note that the requirements on the activation function only applies to the hidden layer, the output nodes are always +assumed to be linear, so as to not restrict the range of output values. + +

    +We note that this theorem is only applicable to a NN with one hidden layer. +Therefore, we can easily construct an NN +that employs activation functions which do not satisfy the above requirements, as long as we have at least one layer +with activation functions that do. Furthermore, although the universal approximation theorem +lays the theoretical foundation for regression with neural networks, it does not say anything about how things work in practice: +A neural network can still be able to approximate a given function reasonably well without having the flexibility to fit all other +functions. + +

    +









    + +

    Mathematical model

    $$ \begin{equation} @@ -247,13 +290,13 @@ $$ $$ In an FFNN of such neurons, the inputs \( x_i \) -are the outputs of the neurons in the preceding layer. Furthermore, a MLP is fully-connected, +are the outputs of the neurons in the preceding layer. Furthermore, an MLP is fully-connected, which means that each neuron receives a weighted sum of the outputs of all neurons in the previous layer.











    -

    Mathematical model

    +

    Mathematical model

    First, for each node \( i \) in the first hidden layer, we calculate a weighted sum \( u_i^1 \) of the input coordinates \( x_j \), @@ -290,7 +333,7 @@ the values of the subsequent layer can be calculated and so forth until the outp











    -

    Mathematical model

    +

    Mathematical model

    The output of neuron \( i \) in layer 2 is thus, @@ -319,7 +362,7 @@ $$











    -

    Mathematical model

    +

    Mathematical model

    We can generalize this expression to an MLP with \( l \) hidden layers. The complete functional form @@ -339,7 +382,7 @@ which illustrates a basic property of MLPs: The only independent variables are t











    -

    Mathematical model

    +

    Mathematical model

    This confirms that an MLP, @@ -361,19 +404,12 @@ $$ where the parameters \( c_i \) are weights and biases. By adjusting these parameters, the activation functions can be shifted up and down or left and right, change slope or be rescaled -which is the key to the flexibility of a NN. - -$$ -\begin{equation} - f_o = f(u_o) = u_o -\label{outputActivation} -\end{equation} -$$ +which is the key to the flexibility of a neural network.











    -

    Matrix-vector notation

    +

    Matrix-vector notation

    We can introduce a more convenient notation for the activations in a NN. @@ -386,7 +422,6 @@ is the bias \( b_i^l \) and activation \( y_i^l \) of node \( i \) in layer \( l We have that \( \mathrm{W}_l \) is a \( N_{l-1} \times N_l \) matrix, while \( \vec{b}_l \) and \( \vec{y}_l \) are \( N_l \times 1 \) column vectors. With this notation, the sum in becomes a matrix-vector multiplication, and we can write the equation for the activations of hidden layer 2 in - $$ \begin{equation} \vec{y}_2 = f_2(\mathrm{W}_2 \vec{y}_{1} + \vec{b}_{2}) = @@ -404,27 +439,87 @@ $$ b^2_1 \\ b^2_2 \\ b^2_3 \\ - \end{array}\right]\right) + \end{array}\right]\right). \label{_auto6} \end{equation} $$ -and we see that the activation of node \( i \) in layer 2 is +

    +









    + +

    Matrix-vector notation and activation

    + +The activation of node \( i \) in layer 2 is $$ \begin{equation} y^2_i = f_2\Bigr(w^2_{i1}y^1_1 + w^2_{i2}y^1_2 + w^2_{i3}y^1_3 + b^2_i\Bigr) = - f_2\left(\sum_{j=1}^3 w^2_{ij} y_j^1 + b^2_i\right) + f_2\left(\sum_{j=1}^3 w^2_{ij} y_j^1 + b^2_i\right). \label{_auto7} \end{equation} $$ -which is in accordance with. Note that This is not just a convenient and compact notation, but also a useful and intuitive way to think about MLPs: The output is calculated by a series of matrix-vector multiplications and vector additions that are used as input to the activation functions. For each operation \( \mathrm{W}_l \vec{y}_{l-1} \) we move forward one layer. +

    +









    + +

    Activation functions

    + +

    +A property that characterizes a neural network, other than its connectivity, is the choice of activation function(s). +As described in, the following restrictions are imposed on an activation function for a FFNN +to fulfill the universal approximation theorem + +

      +
    • Non-constant
    • +
    • Bounded
    • +
    • Monotonically-increasing
    • +
    • Continuous
    • +
    + +









    + +

    Activation functions, Logistic and Hyperbolic ones

    + +

    +The second requirement excludes all linear functions. Furthermore, in a MLP with only linear activation functions, each +layer simply performs a linear transformation of its inputs. + +

    +Regardless of the number of layers, +the output of the NN will be nothing but a linear function of the inputs. Thus we need to introduce some kind of +non-linearity to the NN to be able to fit non-linear functions +Typical examples are the logistic Sigmoid +$$ +\begin{equation} + f(x) = \frac{1}{1 + e^{-x}}, +\label{sigmoidActivationFunction} +\end{equation} +$$ + +and the hyperbolic tangent function +$$ +\begin{equation} + f(x) = \tanh(x) +\label{tanhActivationFunction} +\end{equation} +$$ + +

    +









    + +

    Relevance

    + +The sigmoid function are more biologically plausible because +the output of inactive neurons are zero. Such activation function are called one-sided. However, +it has been shown that the hyperbolic tangent +performs better than the sigmoid for training MLPs. +has become the most popular for deep neural networks +

    diff --git a/doc/pub/NeuralNet/ipynb/NeuralNet.ipynb b/doc/pub/NeuralNet/ipynb/NeuralNet.ipynb index 3b424cdc9..ab5655a33 100644 --- a/doc/pub/NeuralNet/ipynb/NeuralNet.ipynb +++ b/doc/pub/NeuralNet/ipynb/NeuralNet.ipynb @@ -142,6 +142,30 @@ "Other types of NNs could also be mentioned, but are outside the scope of this work. We will now move on to a detailed description\n", "of how a fully-connected FFNN works, and how it can be used to interpolate data sets. \n", "\n", + "## Multilayer perceptrons\n", + "\n", + "One use often so-called fully-connected feed-forward neural networks with three\n", + "or more layers (an input layer, one or more hidden layers and an output layer)\n", + "consisting of neurons that have non-linear activation functions.\n", + "\n", + "Such networks are often called *multilayer perceptrons* (MLPs)\n", + "\n", + "## Why multilayer perceptrons?\n", + "\n", + "According to the *Universal approximation theorem*, a feed-forward neural network with just a single hidden layer containing \n", + "a finite number of neurons can approximate a continuous multidimensional function to arbitrary accuracy, \n", + "assuming the activation function for the hidden layer is a **non-constant, bounded and monotonically-increasing continuous function**.\n", + "Note that the requirements on the activation function only applies to the hidden layer, the output nodes are always\n", + "assumed to be linear, so as to not restrict the range of output values. \n", + "\n", + "We note that this theorem is only applicable to a NN with *one* hidden layer. \n", + "Therefore, we can easily construct an NN \n", + "that employs activation functions which do not satisfy the above requirements, as long as we have at least one layer\n", + "with activation functions that *do*. Furthermore, although the universal approximation theorem\n", + "lays the theoretical foundation for regression with neural networks, it does not say anything about how things work in practice: \n", + "A neural network can still be able to approximate a given function reasonably well without having the flexibility to fit *all other*\n", + "functions. \n", + "\n", "\n", "\n", "## Mathematical model" @@ -167,7 +191,7 @@ "metadata": {}, "source": [ "In an FFNN of such neurons, the *inputs* $x_i$\n", - "are the *outputs* of the neurons in the preceding layer. Furthermore, a MLP is fully-connected, \n", + "are the *outputs* of the neurons in the preceding layer. Furthermore, an MLP is fully-connected, \n", "which means that each neuron receives a weighted sum of the outputs of *all* neurons in the previous layer. \n", "\n", "## Mathematical model\n", @@ -384,28 +408,8 @@ "source": [ "where the parameters $c_i$ are weights and biases. By adjusting these parameters, the activation functions\n", "can be shifted up and down or left and right, change slope or be rescaled \n", - "which is the key to the flexibility of a NN." - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "

    \n", + "which is the key to the flexibility of a neural network. \n", "\n", - "$$\n", - "\\begin{equation}\n", - " f_o = f(u_o) = u_o\n", - "\\label{outputActivation} \\tag{12}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ "### Matrix-vector notation\n", "\n", "We can introduce a more convenient notation for the activations in a NN. \n", @@ -443,8 +447,8 @@ " b^2_1 \\\\\n", " b^2_2 \\\\\n", " b^2_3 \\\\\n", - " \\end{array}\\right]\\right)\n", - "\\label{_auto6} \\tag{13}\n", + " \\end{array}\\right]\\right).\n", + "\\label{_auto6} \\tag{12}\n", "\\end{equation}\n", "$$" ] @@ -453,7 +457,9 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "and we see that the activation of node $i$ in layer 2 is" + "### Matrix-vector notation and activation\n", + "\n", + "The activation of node $i$ in layer 2 is" ] }, { @@ -466,8 +472,8 @@ "$$\n", "\\begin{equation}\n", " y^2_i = f_2\\Bigr(w^2_{i1}y^1_1 + w^2_{i2}y^1_2 + w^2_{i3}y^1_3 + b^2_i\\Bigr) = \n", - " f_2\\left(\\sum_{j=1}^3 w^2_{ij} y_j^1 + b^2_i\\right)\n", - "\\label{_auto7} \\tag{14}\n", + " f_2\\left(\\sum_{j=1}^3 w^2_{ij} y_j^1 + b^2_i\\right).\n", + "\\label{_auto7} \\tag{13}\n", "\\end{equation}\n", "$$" ] @@ -476,11 +482,85 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "which is in accordance with. Note that\n", "This is not just a convenient and compact notation, but also \n", "a useful and intuitive way to think about MLPs: The output is calculated by a series of matrix-vector multiplications\n", "and vector additions that are used as input to the activation functions. For each operation \n", - "$\\mathrm{W}_l \\vec{y}_{l-1}$ we move forward one layer." + "$\\mathrm{W}_l \\vec{y}_{l-1}$ we move forward one layer. \n", + "\n", + "\n", + "### Activation functions\n", + "\n", + "A property that characterizes a neural network, other than its connectivity, is the choice of activation function(s). \n", + "As described in, the following restrictions are imposed on an activation function for a FFNN\n", + "to fulfill the universal approximation theorem\n", + "\n", + " * Non-constant\n", + "\n", + " * Bounded\n", + "\n", + " * Monotonically-increasing\n", + "\n", + " * Continuous\n", + "\n", + "### Activation functions, Logistic and Hyperbolic ones\n", + "\n", + "The second requirement excludes all linear functions. Furthermore, in a MLP with only linear activation functions, each \n", + "layer simply performs a linear transformation of its inputs.\n", + "\n", + "Regardless of the number of layers, \n", + "the output of the NN will be nothing but a linear function of the inputs. Thus we need to introduce some kind of \n", + "non-linearity to the NN to be able to fit non-linear functions\n", + "Typical examples are the logistic *Sigmoid*" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
    \n", + "\n", + "$$\n", + "\\begin{equation}\n", + " f(x) = \\frac{1}{1 + e^{-x}},\n", + "\\label{sigmoidActivationFunction} \\tag{14}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "and the *hyperbolic tangent* function" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
    \n", + "\n", + "$$\n", + "\\begin{equation}\n", + " f(x) = \\tanh(x)\n", + "\\label{tanhActivationFunction} \\tag{15}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Relevance\n", + "\n", + "The *sigmoid* function are more biologically plausible because \n", + "the output of inactive neurons are zero. Such activation function are called *one-sided*. However,\n", + "it has been shown that the hyperbolic tangent \n", + "performs better than the sigmoid for training MLPs. \n", + "has become the most popular for *deep neural networks*" ] } ], diff --git a/doc/pub/NeuralNet/ipynb/ipynb-NeuralNet-src.tar.gz b/doc/pub/NeuralNet/ipynb/ipynb-NeuralNet-src.tar.gz index fa85de6a26982e44208fbaf4b9ac77f1b6c7e88d..69622dcc5a92717e05de1f9e26a1fcab45f41cfd 100644 GIT binary patch literal 212 zcmb2|=3waW6OCeEetZ7B7qg+j@sIOG z%G@_QZP9h>m#1fR`yLN&6BpmaegB;7@ol%|zs1B&JpV^|9^3q7KB4)0o$d0l>A%*j2|2av(zaK>mjz$k zv@SZfBkuhY{oqq-a{iL>Yy7X(TbIk+&a1ysTFvgl*mwr$(CZQK50Ofu0II}=XqOl;e>JxM0kn{)0x_5Sz1SG#K0UaPxy z)!x

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type of NN due the unusual activation functions. Other types of NNs could also be mentioned, but are outside the scope of this work. We will now move on to a detailed description of how a fully-connected FFNN works, and how it can be used to interpolate data sets. +!split +===== Multilayer perceptrons ===== + +One use often so-called fully-connected feed-forward neural networks with three +or more layers (an input layer, one or more hidden layers and an output layer) +consisting of neurons that have non-linear activation functions. + +Such networks are often called *multilayer perceptrons* (MLPs) + +!split +===== Why multilayer perceptrons? ===== + +According to the *Universal approximation theorem*, a feed-forward neural network with just a single hidden layer containing +a finite number of neurons can approximate a continuous multidimensional function to arbitrary accuracy, +assuming the activation function for the hidden layer is a _non-constant, bounded and monotonically-increasing continuous function_. +Note that the requirements on the activation function only applies to the hidden layer, the output nodes are always +assumed to be linear, so as to not restrict the range of output values. + +We note that this theorem is only applicable to a NN with *one* hidden layer. +Therefore, we can easily construct an NN +that employs activation functions which do not satisfy the above requirements, as long as we have at least one layer +with activation functions that *do*. Furthermore, although the universal approximation theorem +lays the theoretical foundation for regression with neural networks, it does not say anything about how things work in practice: +A neural network can still be able to approximate a given function reasonably well without having the flexibility to fit *all other* +functions. + !split @@ -134,7 +160,7 @@ of how a fully-connected FFNN works, and how it can be used to interpolate data \end{equation} !et In an FFNN of such neurons, the *inputs* $x_i$ -are the *outputs* of the neurons in the preceding layer. Furthermore, a MLP is fully-connected, +are the *outputs* of the neurons in the preceding layer. Furthermore, an MLP is fully-connected, which means that each neuron receives a weighted sum of the outputs of *all* neurons in the previous layer. !split @@ -226,16 +252,7 @@ the expression is essentially a nested sum of scaled activation functions of th !et where the parameters $c_i$ are weights and biases. By adjusting these parameters, the activation functions can be shifted up and down or left and right, change slope or be rescaled -which is the key to the flexibility of a NN. - - -!bt -\begin{equation} - f_o = f(u_o) = u_o - label{outputActivation} -\end{equation} -!et - +which is the key to the flexibility of a neural network. !split === Matrix-vector notation === @@ -248,7 +265,6 @@ is the bias $b_i^l$ and activation $y_i^l$ of node $i$ in layer $l$ respectively We have that $\mathrm{W}_l$ is a $N_{l-1} \times N_l$ matrix, while $\vec{b}_l$ and $\vec{y}_l$ are $N_l \times 1$ column vectors. With this notation, the sum in becomes a matrix-vector multiplication, and we can write the equation for the activations of hidden layer 2 in - !bt \begin{equation} \vec{y}_2 = f_2(\mathrm{W}_2 \vec{y}_{1} + \vec{b}_{2}) = @@ -266,21 +282,72 @@ the equation for the activations of hidden layer 2 in b^2_1 \\ b^2_2 \\ b^2_3 \\ - \end{array}\right]\right) + \end{array}\right]\right). \end{equation} !et -and we see that the activation of node $i$ in layer 2 is + +!split +=== Matrix-vector notation and activation === +The activation of node $i$ in layer 2 is !bt \begin{equation} y^2_i = f_2\Bigr(w^2_{i1}y^1_1 + w^2_{i2}y^1_2 + w^2_{i3}y^1_3 + b^2_i\Bigr) = - f_2\left(\sum_{j=1}^3 w^2_{ij} y_j^1 + b^2_i\right) + f_2\left(\sum_{j=1}^3 w^2_{ij} y_j^1 + b^2_i\right). \end{equation} !et -which is in accordance with. Note that This is not just a convenient and compact notation, but also a useful and intuitive way to think about MLPs: The output is calculated by a series of matrix-vector multiplications and vector additions that are used as input to the activation functions. For each operation $\mathrm{W}_l \vec{y}_{l-1}$ we move forward one layer. +!split +=== Activation functions === + + +A property that characterizes a neural network, other than its connectivity, is the choice of activation function(s). +As described in, the following restrictions are imposed on an activation function for a FFNN +to fulfill the universal approximation theorem + + * Non-constant + + * Bounded + + * Monotonically-increasing + + * Continuous + +!split +=== Activation functions, Logistic and Hyperbolic ones === + +The second requirement excludes all linear functions. Furthermore, in a MLP with only linear activation functions, each +layer simply performs a linear transformation of its inputs. + +Regardless of the number of layers, +the output of the NN will be nothing but a linear function of the inputs. Thus we need to introduce some kind of +non-linearity to the NN to be able to fit non-linear functions +Typical examples are the logistic *Sigmoid* +!bt +\begin{equation} + f(x) = \frac{1}{1 + e^{-x}}, + label{sigmoidActivationFunction} +\end{equation} +!et +and the *hyperbolic tangent* function +!bt +\begin{equation} + f(x) = \tanh(x) + label{tanhActivationFunction} +\end{equation} +!et + +!split +=== Relevance === +The *sigmoid* function are more biologically plausible because +the output of inactive neurons are zero. Such activation function are called *one-sided*. However, +it has been shown that the hyperbolic tangent +performs better than the sigmoid for training MLPs. +has become the most popular for *deep neural networks* + +