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doc/src/week1/week1.pdf delete mode 100644 doc/src/week1/week1.snm delete mode 100644 doc/src/week1/week1.tex delete mode 100644 doc/src/week1/week1.toc delete mode 100644 doc/src/week1/week1.vrb diff --git a/doc/pub/week1/html/week1-bs.html b/doc/pub/week1/html/week1-bs.html index e3c1837b..c73b910b 100644 --- a/doc/pub/week1/html/week1-bs.html +++ b/doc/pub/week1/html/week1-bs.html @@ -236,10 +236,28 @@ 2, None, 'other-important-mathematical-operations'), + ('Further mathematical notations', + 2, + None, + 'further-mathematical-notations'), ('Setting up the basic equations for neural networks', 2, None, - 'setting-up-the-basic-equations-for-neural-networks')]} + 'setting-up-the-basic-equations-for-neural-networks'), + ('Overarching view of a neural network', + 2, + None, + 'overarching-view-of-a-neural-network'), + ('The optimzation problem', 2, None, 'the-optimzation-problem'), + ('Other ingredients of a neural network', + 2, + None, + 'other-ingredients-of-a-neural-network'), + ('Other parameters', 2, None, 'other-parameters'), + ('Setting up the equations for a neural network', + 2, + None, + 'setting-up-the-equations-for-a-neural-network')]} end of tocinfo --> @@ -335,7 +353,13 @@
  • Vector-matrix and Matrix-matrix multiplication
  • Important Mathematical Operations
  • Other important mathematical operations
    • +
  • Further mathematical notations
  • Setting up the basic equations for neural networks
    • +
  • Overarching view of a neural network
    • +
  • The optimzation problem
    • +
  • Other ingredients of a neural network
    • +
  • Other parameters
    • +
  • Setting up the equations for a neural network
    • @@ -1058,21 +1082,20 @@

    Representing the wave fun

    $$ - F_{rbm}(\mathbf{x},\mathbf{h}) = \frac{1}{Z} e^{-\frac{1}{T_0}E(\mathbf{x},\mathbf{h})}. + P_{rbm}(\mathbf{x},\mathbf{h}) = \frac{1}{Z} e^{-\frac{1}{T_0}E(\mathbf{x},\mathbf{h})}. $$

    To find the marginal distribution of \( \boldsymbol{x} \) we set:

    $$ - F_{rbm}(\mathbf{x}) =\frac{1}{Z}\sum_\mathbf{h} e^{-E(\mathbf{x}, \mathbf{h})}. + P_{rbm}(\mathbf{x}) =\frac{1}{Z}\sum_\mathbf{h} e^{-E(\mathbf{x}, \mathbf{h})}. $$

    Now this is what we use to represent the wave function, calling it a neural-network quantum state (NQS)

    $$ - \Psi (\mathbf{X}) = F_{rbm}(\mathbf{x}), + \vert\Psi (\mathbf{X})\vert^2 = P_{rbm}(\mathbf{x}). $$ -

    or we could square the wave function.

    Define the cost function

    @@ -1232,20 +1255,21 @@

    Basic Matrix Features

    or in terms of its column vectors \( \boldsymbol{a}_i \) as

    $$ - \mathbf{A} = + \boldsymbol{A} = \begin{bmatrix}\boldsymbol{a}_{0} & \boldsymbol{a}_{1} & \boldsymbol{a}_{2} & \dots & \dots & \boldsymbol{a}_{n-2} & \boldsymbol{a}_{n-1}\end{bmatrix}. $$ +

    We can think of a matrix as a diagram of in general \( n \) rowns and \( m \) columns. In the example here we have a square matrix.

    The inverse of a matrix

    -

    The inverse of a matrix (if it exists) is defined by

    +

    The inverse of a square matrix (if it exists) is defined by

    $$ -\mathbf{A}^{-1} \cdot \mathbf{A} = I, +\boldsymbol{A}^{-1} \cdot \boldsymbol{A} = I, $$

    where \( \boldsymbol{I} \) is the unit matrix.

    @@ -1300,15 +1324,15 @@

    Matrix Features

    -

    For an \( N\times N \) matrix \( \mathbf{A} \) the following properties are all equivalent

    +

    For an \( n\times n \) matrix \( \boldsymbol{A} \) the following properties are all equivalent

      -
    • If the inverse of \( \mathbf{A} \) exists, \( \mathbf{A} \) is nonsingular.
      • -
    • The equation \( \mathbf{Ax}=0 \) implies \( \mathbf{x}=0 \).
      • -
    • The rows of \( \mathbf{A} \) form a basis of \( R^N \).
      • -
    • The columns of \( \mathbf{A} \) form a basis of \( R^N \).
      • -
    • \( \mathbf{A} \) is a product of elementary matrices.
      • -
    • \( 0 \) is not eigenvalue of \( \mathbf{A} \).
      • +
    • If the inverse of \( \boldsymbol{A} \) exists, \( \boldsymbol{A} \) is nonsingular.
      • +
    • The equation \( \boldsymbol{Ax}=0 \) implies \( \boldsymbol{x}=0 \).
      • +
    • The rows of \( \boldsymbol{A} \) form a basis of \( R^N \).
      • +
    • The columns of \( \boldsymbol{A} \) form a basis of \( R^N \).
      • +
    • \( \boldsymbol{A} \) is a product of elementary matrices.
      • +
    • \( 0 \) is not an eigenvalue of \( \boldsymbol{A} \).
    @@ -1320,13 +1344,13 @@

    Important Mathematical

    The basic matrix operations that we will deal with are addition and subtraction

    $$ -\mathbf{A}= \mathbf{B}\pm\mathbf{C} \Longrightarrow a_{ij} = b_{ij}\pm c_{ij}, +\boldsymbol{A}= \boldsymbol{B}\pm\boldsymbol{C} \Longrightarrow a_{ij} = b_{ij}\pm c_{ij}, $$

    and scalar-matrix multiplication

    $$ -\mathbf{A}= \gamma\mathbf{B} \Longrightarrow a_{ij} = \gamma b_{ij}. +\boldsymbol{A}= \gamma\boldsymbol{B} \Longrightarrow a_{ij} = \gamma b_{ij}. $$ @@ -1335,19 +1359,19 @@

    Vector-ma

    We have also vector-matrix multiplications

    $$ -\mathbf{y}=\mathbf{Ax} \Longrightarrow y_{i} = \sum_{j=1}^{n} a_{ij}x_j, +\boldsymbol{y}=\boldsymbol{Ax} \Longrightarrow y_{i} = \sum_{j=1}^{n} a_{ij}x_j, $$

    and matrix-matrix multiplications

    $$ -\mathbf{A}=\mathbf{BC} \Longrightarrow a_{ij} = \sum_{k=1}^{n} b_{ik}c_{kj}, +\boldsymbol{A}=\boldsymbol{BC} \Longrightarrow a_{ij} = \sum_{k=1}^{n} b_{ik}c_{kj}, $$

    and transpositions of a matrix

    $$ -\mathbf{A}=\mathbf{B}^T \Longrightarrow a_{ij} = b_{ji}. +\boldsymbol{A}=\boldsymbol{B}^T \Longrightarrow a_{ij} = b_{ji}. $$ @@ -1357,13 +1381,13 @@

    Important Mathematical

    Similarly, important vector operations that we will deal with are addition and subtraction

    $$ -\mathbf{x}= \mathbf{y}\pm\mathbf{z} \Longrightarrow x_{i} = y_{i}\pm z_{i}, +\boldsymbol{x}= \boldsymbol{y}\pm\boldsymbol{z} \Longrightarrow x_{i} = y_{i}\pm z_{i}, $$

    scalar-vector multiplication

    $$ -\mathbf{x}= \gamma\mathbf{y} \Longrightarrow x_{i} = \gamma y_{i}, +\boldsymbol{x}= \gamma\boldsymbol{y} \Longrightarrow x_{i} = \gamma y_{i}, $$ @@ -1371,32 +1395,127 @@

    Important Mathematical

    Other important mathematical operations

    and vector-vector multiplication (called Hadamard multiplication)

    $$ -\mathbf{x}=\mathbf{yz} \Longrightarrow x_{i} = y_{i}z_i. +\boldsymbol{x}=\boldsymbol{yz} \Longrightarrow x_{i} = y_{i}z_i. $$

    Finally, as already metnioned, the inner or so-called dot product resulting in a constant

    $$ -x=\mathbf{y}^T\mathbf{z} \Longrightarrow x = \sum_{j=1}^{n} y_{j}z_{j}, +x=\boldsymbol{y}^T\boldsymbol{z} \Longrightarrow x = \sum_{j=1}^{n} y_{j}z_{j}, $$

    and the outer product, which yields a matrix,

    $$ -\mathbf{A}= \mathbf{yz}^T \Longrightarrow a_{ij} = y_{i}z_{j}, +\boldsymbol{A}= \boldsymbol{y}\boldsymbol{z}^T \Longrightarrow a_{ij} = y_{i}z_{j}, $$ + +

    Further mathematical notations

    +
      +
    1. For all/any \( \forall \)
      2. +
    2. Implies \( \implies \)
      3. +
    3. Equivalent \( \equiv \)
      4. +
    4. Real variable \( \mathbb{R} \)
      5. +
    5. Integer variable \( \mathbb{I} \)
      6. +
    6. Complex variable \( \mathbb{C} \)
      7. +

    Setting up the basic equations for neural networks

    -

    Neural networks, its so-called feed-forward form, where each +

    Neural networks, in its so-called feed-forward form, where each iterations contains a feed-forward stage and a back-propgagation stage, consist of series of affine matrix-matrix and matrix-vector multiplications. The unknown parameters (the so-called biases and -weights which deternine the architecture of a neural network), are uptaded iteratively +weights which deternine the architecture of a neural network), are +uptaded iteratively using the so-called back-propagation algorithm. +This algorithm corresponds to the so-called reverse mode of the +automatic differentation algorithm. These algorithms will be discussed +in more detail below.

    +

    We start however first with the definitions of the various variables which make up a neural network.

    + + +

    Overarching view of a neural network

    + +

    The architecture of a neural network defines our model. This model +aims at describing some function \( f(\boldsymbol{x} \) which aims at describing +some final result (outputs or tagrget values) given a specific inpput +\( \boldsymbol{x} \). Note that here \( \boldsymbol{y} \) and \( \boldsymbol{x} \) are not limited to be +vectors. +

    + +

    The architecture consists of

    +
      +
    1. An input and an output layer where the input layer is defined by the inputs \( \boldsymbol{x} \). The output layer produces the model ouput \( \boldsymbol{\tilde{y}} \) which is compared with the target value \( \boldsymbol{y} \)
      2. +
    2. A given number of hidden layers and neurons/nodes/units for each layer (this may vary)
      3. +
    3. A given activation function \( \sigma(\boldsymbol{z}) \) with arguments \( \boldsymbol{z} \) to be defined below. The activation functions may differ from layer to layer.
      4. +
    4. The last layer, normally called output layer has normally an activation function tailored to the specific problem
      5. +
    5. Finally we define a so-called cost or loss function which is used to gauge the quality of our model.
      6. +
    + +

    The optimzation problem

    + +

    The cost function is a function of the unknown parameters +\( \boldsymbol{\Theta} \) where the latter is a container for all possible +parameters needed to define a neural network +

    + +

    If we are dealing with a regression task a typical cost/loss function +is the mean squared error +

    +$$ +C(\boldsymbol{\Theta})=\frac{1}{n}\left\{\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\theta}\right)^T\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\theta}\right)\right\}. +$$ + +

    This function represents one of many possible ways to define +the so-called cost function. +

    + +

    For neural networks the parameters +\( \boldsymbol{\Theta} \) are given by the so-called weights and biases (to be +defined below). +

    + +

    The weights are given by matrix elements \( w_{ij}^{(l)} \) where the +superscript indicates the layer number. The biases are typically given +by vector elements representing each single node of a given layer, +that is \( b_j^{(l)} \). +

    + + +

    Other ingredients of a neural network

    + +

    Having defined the architecture of a neural network, the optimization +of the cost function with respect to the parameters \( \boldsymbol{\Theta} \), +involves the calculations of gradients and their optimization. The +gradients represent the derivatives of a multidimensional object and +are often approximated by various gradient methods, including +

    +
      +
    1. various quasi-Newton methods,
      2. +
    2. plain gradient descent (GD) with a constant learning rate \( \eta \),
      3. +
    3. GD with momentum and other approximations to the learning rates such as
      4. +
        +
      • Adapative gradient (ADAgrad)
        • +
      • Root mean-square propagation (RMSprop)
        • +
      • Adaptive gradient with momentum (ADAM) and many other
        • +
      +
    4. Stochastic gradient descent and various families of learning rate approximations
      5. +
    + +

    Other parameters

    + +

    In addition to the above, there are often additional hyperparamaters +which are included in the setup of a neural network. These will be +discussed below. +

    + + +

    Setting up the equations for a neural network

    +
    diff --git a/doc/pub/week1/html/week1-reveal.html b/doc/pub/week1/html/week1-reveal.html index debf0eea..e46b59a9 100644 --- a/doc/pub/week1/html/week1-reveal.html +++ b/doc/pub/week1/html/week1-reveal.html @@ -957,7 +957,7 @@

    Representing the wave function

     
    $$ - F_{rbm}(\mathbf{x},\mathbf{h}) = \frac{1}{Z} e^{-\frac{1}{T_0}E(\mathbf{x},\mathbf{h})}. + P_{rbm}(\mathbf{x},\mathbf{h}) = \frac{1}{Z} e^{-\frac{1}{T_0}E(\mathbf{x},\mathbf{h})}. $$

     
    @@ -965,18 +965,16 @@

    Representing the wave function

     
    $$ - F_{rbm}(\mathbf{x}) =\frac{1}{Z}\sum_\mathbf{h} e^{-E(\mathbf{x}, \mathbf{h})}. + P_{rbm}(\mathbf{x}) =\frac{1}{Z}\sum_\mathbf{h} e^{-E(\mathbf{x}, \mathbf{h})}. $$

     

    Now this is what we use to represent the wave function, calling it a neural-network quantum state (NQS)

     
    $$ - \Psi (\mathbf{X}) = F_{rbm}(\mathbf{x}), + \vert\Psi (\mathbf{X})\vert^2 = P_{rbm}(\mathbf{x}). $$

     
    - -

    or we could square the wave function.

    @@ -1162,10 +1160,12 @@

    Basic Matrix Features

    or in terms of its column vectors \( \boldsymbol{a}_i \) as

     
    $$ - \mathbf{A} = + \boldsymbol{A} = \begin{bmatrix}\boldsymbol{a}_{0} & \boldsymbol{a}_{1} & \boldsymbol{a}_{2} & \dots & \dots & \boldsymbol{a}_{n-2} & \boldsymbol{a}_{n-1}\end{bmatrix}. $$

     
    + +

    We can think of a matrix as a diagram of in general \( n \) rowns and \( m \) columns. In the example here we have a square matrix.

    @@ -1173,11 +1173,11 @@

    The inverse of a matrix

    -

    The inverse of a matrix (if it exists) is defined by

    +

    The inverse of a square matrix (if it exists) is defined by

     
    $$ -\mathbf{A}^{-1} \cdot \mathbf{A} = I, +\boldsymbol{A}^{-1} \cdot \boldsymbol{A} = I, $$

     
    @@ -1238,21 +1238,21 @@

    Matrix Features

    Some Equivalent Statements

    -

    For an \( N\times N \) matrix \( \mathbf{A} \) the following properties are all equivalent

    +

    For an \( n\times n \) matrix \( \boldsymbol{A} \) the following properties are all equivalent

      -

    • If the inverse of \( \mathbf{A} \) exists, \( \mathbf{A} \) is nonsingular.
      • +

    • If the inverse of \( \boldsymbol{A} \) exists, \( \boldsymbol{A} \) is nonsingular.
      • -

    • The equation \( \mathbf{Ax}=0 \) implies \( \mathbf{x}=0 \).
      • +

    • The equation \( \boldsymbol{Ax}=0 \) implies \( \boldsymbol{x}=0 \).
      • -

    • The rows of \( \mathbf{A} \) form a basis of \( R^N \).
      • +

    • The rows of \( \boldsymbol{A} \) form a basis of \( R^N \).
      • -

    • The columns of \( \mathbf{A} \) form a basis of \( R^N \).
      • +

    • The columns of \( \boldsymbol{A} \) form a basis of \( R^N \).
      • -

    • \( \mathbf{A} \) is a product of elementary matrices.
      • +

    • \( \boldsymbol{A} \) is a product of elementary matrices.
      • -

    • \( 0 \) is not eigenvalue of \( \mathbf{A} \).
      • +

    • \( 0 \) is not an eigenvalue of \( \boldsymbol{A} \).
    @@ -1264,7 +1264,7 @@

    Important Mathematical Operations  
    $$ -\mathbf{A}= \mathbf{B}\pm\mathbf{C} \Longrightarrow a_{ij} = b_{ij}\pm c_{ij}, +\boldsymbol{A}= \boldsymbol{B}\pm\boldsymbol{C} \Longrightarrow a_{ij} = b_{ij}\pm c_{ij}, $$

     
    @@ -1272,7 +1272,7 @@

    Important Mathematical Operations  
    $$ -\mathbf{A}= \gamma\mathbf{B} \Longrightarrow a_{ij} = \gamma b_{ij}. +\boldsymbol{A}= \gamma\boldsymbol{B} \Longrightarrow a_{ij} = \gamma b_{ij}. $$

     
    @@ -1283,7 +1283,7 @@

    Vector-matrix and Matrix

    We have also vector-matrix multiplications

     
    $$ -\mathbf{y}=\mathbf{Ax} \Longrightarrow y_{i} = \sum_{j=1}^{n} a_{ij}x_j, +\boldsymbol{y}=\boldsymbol{Ax} \Longrightarrow y_{i} = \sum_{j=1}^{n} a_{ij}x_j, $$

     
    @@ -1291,7 +1291,7 @@

    Vector-matrix and Matrix

     
    $$ -\mathbf{A}=\mathbf{BC} \Longrightarrow a_{ij} = \sum_{k=1}^{n} b_{ik}c_{kj}, +\boldsymbol{A}=\boldsymbol{BC} \Longrightarrow a_{ij} = \sum_{k=1}^{n} b_{ik}c_{kj}, $$

     
    @@ -1299,7 +1299,7 @@

    Vector-matrix and Matrix

     
    $$ -\mathbf{A}=\mathbf{B}^T \Longrightarrow a_{ij} = b_{ji}. +\boldsymbol{A}=\boldsymbol{B}^T \Longrightarrow a_{ij} = b_{ji}. $$

     
    @@ -1311,7 +1311,7 @@

    Important Mathematical Operations  
    $$ -\mathbf{x}= \mathbf{y}\pm\mathbf{z} \Longrightarrow x_{i} = y_{i}\pm z_{i}, +\boldsymbol{x}= \boldsymbol{y}\pm\boldsymbol{z} \Longrightarrow x_{i} = y_{i}\pm z_{i}, $$

     
    @@ -1319,7 +1319,7 @@

    Important Mathematical Operations  
    $$ -\mathbf{x}= \gamma\mathbf{y} \Longrightarrow x_{i} = \gamma y_{i}, +\boldsymbol{x}= \gamma\boldsymbol{y} \Longrightarrow x_{i} = \gamma y_{i}, $$

     
    @@ -1329,7 +1329,7 @@

    Other important mathematical op

    and vector-vector multiplication (called Hadamard multiplication)

     
    $$ -\mathbf{x}=\mathbf{yz} \Longrightarrow x_{i} = y_{i}z_i. +\boldsymbol{x}=\boldsymbol{yz} \Longrightarrow x_{i} = y_{i}z_i. $$

     
    @@ -1337,7 +1337,7 @@

    Other important mathematical op

     
    $$ -x=\mathbf{y}^T\mathbf{z} \Longrightarrow x = \sum_{j=1}^{n} y_{j}z_{j}, +x=\boldsymbol{y}^T\boldsymbol{z} \Longrightarrow x = \sum_{j=1}^{n} y_{j}z_{j}, $$

     
    @@ -1345,22 +1345,132 @@

    Other important mathematical op

     
    $$ -\mathbf{A}= \mathbf{yz}^T \Longrightarrow a_{ij} = y_{i}z_{j}, +\boldsymbol{A}= \boldsymbol{y}\boldsymbol{z}^T \Longrightarrow a_{ij} = y_{i}z_{j}, $$

     
    +

    +

    Further mathematical notations

    +
      +

    1. For all/any \( \forall \)
      2. +

    2. Implies \( \implies \)
      3. +

    3. Equivalent \( \equiv \)
      4. +

    4. Real variable \( \mathbb{R} \)
      5. +

    5. Integer variable \( \mathbb{I} \)
      6. +

    6. Complex variable \( \mathbb{C} \)
      7. +
    +
    +

    Setting up the basic equations for neural networks

    -

    Neural networks, its so-called feed-forward form, where each +

    Neural networks, in its so-called feed-forward form, where each iterations contains a feed-forward stage and a back-propgagation stage, consist of series of affine matrix-matrix and matrix-vector multiplications. The unknown parameters (the so-called biases and -weights which deternine the architecture of a neural network), are uptaded iteratively +weights which deternine the architecture of a neural network), are +uptaded iteratively using the so-called back-propagation algorithm. +This algorithm corresponds to the so-called reverse mode of the +automatic differentation algorithm. These algorithms will be discussed +in more detail below. +

    + +

    We start however first with the definitions of the various variables which make up a neural network.

    +
    + +
    +

    Overarching view of a neural network

    + +

    The architecture of a neural network defines our model. This model +aims at describing some function \( f(\boldsymbol{x} \) which aims at describing +some final result (outputs or tagrget values) given a specific inpput +\( \boldsymbol{x} \). Note that here \( \boldsymbol{y} \) and \( \boldsymbol{x} \) are not limited to be +vectors. +

    + +

    The architecture consists of

    +
      +

    1. An input and an output layer where the input layer is defined by the inputs \( \boldsymbol{x} \). The output layer produces the model ouput \( \boldsymbol{\tilde{y}} \) which is compared with the target value \( \boldsymbol{y} \)
      2. +

    2. A given number of hidden layers and neurons/nodes/units for each layer (this may vary)
      3. +

    3. A given activation function \( \sigma(\boldsymbol{z}) \) with arguments \( \boldsymbol{z} \) to be defined below. The activation functions may differ from layer to layer.
      4. +

    4. The last layer, normally called output layer has normally an activation function tailored to the specific problem
      5. +

    5. Finally we define a so-called cost or loss function which is used to gauge the quality of our model.
      6. +
    +
    + +
    +

    The optimzation problem

    + +

    The cost function is a function of the unknown parameters +\( \boldsymbol{\Theta} \) where the latter is a container for all possible +parameters needed to define a neural network +

    + +

    If we are dealing with a regression task a typical cost/loss function +is the mean squared error +

    +

     
    +$$ +C(\boldsymbol{\Theta})=\frac{1}{n}\left\{\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\theta}\right)^T\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\theta}\right)\right\}. +$$ +

     
    + +

    This function represents one of many possible ways to define +the so-called cost function. +

    + +

    For neural networks the parameters +\( \boldsymbol{\Theta} \) are given by the so-called weights and biases (to be +defined below). +

    + +

    The weights are given by matrix elements \( w_{ij}^{(l)} \) where the +superscript indicates the layer number. The biases are typically given +by vector elements representing each single node of a given layer, +that is \( b_j^{(l)} \). +

    +
    + +
    +

    Other ingredients of a neural network

    + +

    Having defined the architecture of a neural network, the optimization +of the cost function with respect to the parameters \( \boldsymbol{\Theta} \), +involves the calculations of gradients and their optimization. The +gradients represent the derivatives of a multidimensional object and +are often approximated by various gradient methods, including +

    +
      +

    1. various quasi-Newton methods,
      2. +

    2. plain gradient descent (GD) with a constant learning rate \( \eta \),
      3. +

    3. GD with momentum and other approximations to the learning rates such as
      4. +
        + +

      • Adapative gradient (ADAgrad)
        • + +

      • Root mean-square propagation (RMSprop)
        • + +

      • Adaptive gradient with momentum (ADAM) and many other
        • +
      +

      +

    4. Stochastic gradient descent and various families of learning rate approximations
      5. +
    +
    + +
    +

    Other parameters

    + +

    In addition to the above, there are often additional hyperparamaters +which are included in the setup of a neural network. These will be +discussed below.

    +
    +

    Setting up the equations for a neural network

    +
    +

    diff --git a/doc/pub/week1/html/week1-solarized.html b/doc/pub/week1/html/week1-solarized.html index d086bb1a..bfde172d 100644 --- a/doc/pub/week1/html/week1-solarized.html +++ b/doc/pub/week1/html/week1-solarized.html @@ -263,10 +263,28 @@ 2, None, 'other-important-mathematical-operations'), + ('Further mathematical notations', + 2, + None, + 'further-mathematical-notations'), ('Setting up the basic equations for neural networks', 2, None, - 'setting-up-the-basic-equations-for-neural-networks')]} + 'setting-up-the-basic-equations-for-neural-networks'), + ('Overarching view of a neural network', + 2, + None, + 'overarching-view-of-a-neural-network'), + ('The optimzation problem', 2, None, 'the-optimzation-problem'), + ('Other ingredients of a neural network', + 2, + None, + 'other-ingredients-of-a-neural-network'), + ('Other parameters', 2, None, 'other-parameters'), + ('Setting up the equations for a neural network', + 2, + None, + 'setting-up-the-equations-for-a-neural-network')]} end of tocinfo --> @@ -979,21 +997,20 @@

    Representing the wave function

    $$ - F_{rbm}(\mathbf{x},\mathbf{h}) = \frac{1}{Z} e^{-\frac{1}{T_0}E(\mathbf{x},\mathbf{h})}. + P_{rbm}(\mathbf{x},\mathbf{h}) = \frac{1}{Z} e^{-\frac{1}{T_0}E(\mathbf{x},\mathbf{h})}. $$

    To find the marginal distribution of \( \boldsymbol{x} \) we set:

    $$ - F_{rbm}(\mathbf{x}) =\frac{1}{Z}\sum_\mathbf{h} e^{-E(\mathbf{x}, \mathbf{h})}. + P_{rbm}(\mathbf{x}) =\frac{1}{Z}\sum_\mathbf{h} e^{-E(\mathbf{x}, \mathbf{h})}. $$

    Now this is what we use to represent the wave function, calling it a neural-network quantum state (NQS)

    $$ - \Psi (\mathbf{X}) = F_{rbm}(\mathbf{x}), + \vert\Psi (\mathbf{X})\vert^2 = P_{rbm}(\mathbf{x}). $$ -

    or we could square the wave function.











    Define the cost function

    @@ -1153,20 +1170,21 @@

    Basic Matrix Features

    or in terms of its column vectors \( \boldsymbol{a}_i \) as

    $$ - \mathbf{A} = + \boldsymbol{A} = \begin{bmatrix}\boldsymbol{a}_{0} & \boldsymbol{a}_{1} & \boldsymbol{a}_{2} & \dots & \dots & \boldsymbol{a}_{n-2} & \boldsymbol{a}_{n-1}\end{bmatrix}. $$ +

    We can think of a matrix as a diagram of in general \( n \) rowns and \( m \) columns. In the example here we have a square matrix.











    The inverse of a matrix

    -

    The inverse of a matrix (if it exists) is defined by

    +

    The inverse of a square matrix (if it exists) is defined by

    $$ -\mathbf{A}^{-1} \cdot \mathbf{A} = I, +\boldsymbol{A}^{-1} \cdot \boldsymbol{A} = I, $$

    where \( \boldsymbol{I} \) is the unit matrix.

    @@ -1215,15 +1233,15 @@

    Matrix Features

    Some Equivalent Statements

    -

    For an \( N\times N \) matrix \( \mathbf{A} \) the following properties are all equivalent

    +

    For an \( n\times n \) matrix \( \boldsymbol{A} \) the following properties are all equivalent

      -
    • If the inverse of \( \mathbf{A} \) exists, \( \mathbf{A} \) is nonsingular.
      • -
    • The equation \( \mathbf{Ax}=0 \) implies \( \mathbf{x}=0 \).
      • -
    • The rows of \( \mathbf{A} \) form a basis of \( R^N \).
      • -
    • The columns of \( \mathbf{A} \) form a basis of \( R^N \).
      • -
    • \( \mathbf{A} \) is a product of elementary matrices.
      • -
    • \( 0 \) is not eigenvalue of \( \mathbf{A} \).
      • +
    • If the inverse of \( \boldsymbol{A} \) exists, \( \boldsymbol{A} \) is nonsingular.
      • +
    • The equation \( \boldsymbol{Ax}=0 \) implies \( \boldsymbol{x}=0 \).
      • +
    • The rows of \( \boldsymbol{A} \) form a basis of \( R^N \).
      • +
    • The columns of \( \boldsymbol{A} \) form a basis of \( R^N \).
      • +
    • \( \boldsymbol{A} \) is a product of elementary matrices.
      • +
    • \( 0 \) is not an eigenvalue of \( \boldsymbol{A} \).
    @@ -1234,13 +1252,13 @@

    Important Mathematical Operations The basic matrix operations that we will deal with are addition and subtraction

    $$ -\mathbf{A}= \mathbf{B}\pm\mathbf{C} \Longrightarrow a_{ij} = b_{ij}\pm c_{ij}, +\boldsymbol{A}= \boldsymbol{B}\pm\boldsymbol{C} \Longrightarrow a_{ij} = b_{ij}\pm c_{ij}, $$

    and scalar-matrix multiplication

    $$ -\mathbf{A}= \gamma\mathbf{B} \Longrightarrow a_{ij} = \gamma b_{ij}. +\boldsymbol{A}= \gamma\boldsymbol{B} \Longrightarrow a_{ij} = \gamma b_{ij}. $$ @@ -1249,19 +1267,19 @@

    Vector-matrix and Matrix

    We have also vector-matrix multiplications

    $$ -\mathbf{y}=\mathbf{Ax} \Longrightarrow y_{i} = \sum_{j=1}^{n} a_{ij}x_j, +\boldsymbol{y}=\boldsymbol{Ax} \Longrightarrow y_{i} = \sum_{j=1}^{n} a_{ij}x_j, $$

    and matrix-matrix multiplications

    $$ -\mathbf{A}=\mathbf{BC} \Longrightarrow a_{ij} = \sum_{k=1}^{n} b_{ik}c_{kj}, +\boldsymbol{A}=\boldsymbol{BC} \Longrightarrow a_{ij} = \sum_{k=1}^{n} b_{ik}c_{kj}, $$

    and transpositions of a matrix

    $$ -\mathbf{A}=\mathbf{B}^T \Longrightarrow a_{ij} = b_{ji}. +\boldsymbol{A}=\boldsymbol{B}^T \Longrightarrow a_{ij} = b_{ji}. $$ @@ -1271,13 +1289,13 @@

    Important Mathematical Operations Similarly, important vector operations that we will deal with are addition and subtraction

    $$ -\mathbf{x}= \mathbf{y}\pm\mathbf{z} \Longrightarrow x_{i} = y_{i}\pm z_{i}, +\boldsymbol{x}= \boldsymbol{y}\pm\boldsymbol{z} \Longrightarrow x_{i} = y_{i}\pm z_{i}, $$

    scalar-vector multiplication

    $$ -\mathbf{x}= \gamma\mathbf{y} \Longrightarrow x_{i} = \gamma y_{i}, +\boldsymbol{x}= \gamma\boldsymbol{y} \Longrightarrow x_{i} = \gamma y_{i}, $$ @@ -1285,32 +1303,127 @@

    Important Mathematical Operations Other important mathematical operations

    and vector-vector multiplication (called Hadamard multiplication)

    $$ -\mathbf{x}=\mathbf{yz} \Longrightarrow x_{i} = y_{i}z_i. +\boldsymbol{x}=\boldsymbol{yz} \Longrightarrow x_{i} = y_{i}z_i. $$

    Finally, as already metnioned, the inner or so-called dot product resulting in a constant

    $$ -x=\mathbf{y}^T\mathbf{z} \Longrightarrow x = \sum_{j=1}^{n} y_{j}z_{j}, +x=\boldsymbol{y}^T\boldsymbol{z} \Longrightarrow x = \sum_{j=1}^{n} y_{j}z_{j}, $$

    and the outer product, which yields a matrix,

    $$ -\mathbf{A}= \mathbf{yz}^T \Longrightarrow a_{ij} = y_{i}z_{j}, +\boldsymbol{A}= \boldsymbol{y}\boldsymbol{z}^T \Longrightarrow a_{ij} = y_{i}z_{j}, $$ +









    +

    Further mathematical notations

    +
      +
    1. For all/any \( \forall \)
      2. +
    2. Implies \( \implies \)
      3. +
    3. Equivalent \( \equiv \)
      4. +
    4. Real variable \( \mathbb{R} \)
      5. +
    5. Integer variable \( \mathbb{I} \)
      6. +
    6. Complex variable \( \mathbb{C} \)
      7. +










    Setting up the basic equations for neural networks

    -

    Neural networks, its so-called feed-forward form, where each +

    Neural networks, in its so-called feed-forward form, where each iterations contains a feed-forward stage and a back-propgagation stage, consist of series of affine matrix-matrix and matrix-vector multiplications. The unknown parameters (the so-called biases and -weights which deternine the architecture of a neural network), are uptaded iteratively +weights which deternine the architecture of a neural network), are +uptaded iteratively using the so-called back-propagation algorithm. +This algorithm corresponds to the so-called reverse mode of the +automatic differentation algorithm. These algorithms will be discussed +in more detail below.

    +

    We start however first with the definitions of the various variables which make up a neural network.

    + +









    +

    Overarching view of a neural network

    + +

    The architecture of a neural network defines our model. This model +aims at describing some function \( f(\boldsymbol{x} \) which aims at describing +some final result (outputs or tagrget values) given a specific inpput +\( \boldsymbol{x} \). Note that here \( \boldsymbol{y} \) and \( \boldsymbol{x} \) are not limited to be +vectors. +

    + +

    The architecture consists of

    +
      +
    1. An input and an output layer where the input layer is defined by the inputs \( \boldsymbol{x} \). The output layer produces the model ouput \( \boldsymbol{\tilde{y}} \) which is compared with the target value \( \boldsymbol{y} \)
      2. +
    2. A given number of hidden layers and neurons/nodes/units for each layer (this may vary)
      3. +
    3. A given activation function \( \sigma(\boldsymbol{z}) \) with arguments \( \boldsymbol{z} \) to be defined below. The activation functions may differ from layer to layer.
      4. +
    4. The last layer, normally called output layer has normally an activation function tailored to the specific problem
      5. +
    5. Finally we define a so-called cost or loss function which is used to gauge the quality of our model.
      6. +
    +









    +

    The optimzation problem

    + +

    The cost function is a function of the unknown parameters +\( \boldsymbol{\Theta} \) where the latter is a container for all possible +parameters needed to define a neural network +

    + +

    If we are dealing with a regression task a typical cost/loss function +is the mean squared error +

    +$$ +C(\boldsymbol{\Theta})=\frac{1}{n}\left\{\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\theta}\right)^T\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\theta}\right)\right\}. +$$ + +

    This function represents one of many possible ways to define +the so-called cost function. +

    + +

    For neural networks the parameters +\( \boldsymbol{\Theta} \) are given by the so-called weights and biases (to be +defined below). +

    + +

    The weights are given by matrix elements \( w_{ij}^{(l)} \) where the +superscript indicates the layer number. The biases are typically given +by vector elements representing each single node of a given layer, +that is \( b_j^{(l)} \). +

    + +









    +

    Other ingredients of a neural network

    + +

    Having defined the architecture of a neural network, the optimization +of the cost function with respect to the parameters \( \boldsymbol{\Theta} \), +involves the calculations of gradients and their optimization. The +gradients represent the derivatives of a multidimensional object and +are often approximated by various gradient methods, including +

    +
      +
    1. various quasi-Newton methods,
      2. +
    2. plain gradient descent (GD) with a constant learning rate \( \eta \),
      3. +
    3. GD with momentum and other approximations to the learning rates such as
      4. +
        +
      • Adapative gradient (ADAgrad)
        • +
      • Root mean-square propagation (RMSprop)
        • +
      • Adaptive gradient with momentum (ADAM) and many other
        • +
      +
    4. Stochastic gradient descent and various families of learning rate approximations
      5. +
    +









    +

    Other parameters

    + +

    In addition to the above, there are often additional hyperparamaters +which are included in the setup of a neural network. These will be +discussed below. +

    + +









    +

    Setting up the equations for a neural network

    +
    © 1999-2024, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license diff --git a/doc/pub/week1/html/week1.html b/doc/pub/week1/html/week1.html index 7b9ac0da..76612115 100644 --- a/doc/pub/week1/html/week1.html +++ b/doc/pub/week1/html/week1.html @@ -340,10 +340,28 @@ 2, None, 'other-important-mathematical-operations'), + ('Further mathematical notations', + 2, + None, + 'further-mathematical-notations'), ('Setting up the basic equations for neural networks', 2, None, - 'setting-up-the-basic-equations-for-neural-networks')]} + 'setting-up-the-basic-equations-for-neural-networks'), + ('Overarching view of a neural network', + 2, + None, + 'overarching-view-of-a-neural-network'), + ('The optimzation problem', 2, None, 'the-optimzation-problem'), + ('Other ingredients of a neural network', + 2, + None, + 'other-ingredients-of-a-neural-network'), + ('Other parameters', 2, None, 'other-parameters'), + ('Setting up the equations for a neural network', + 2, + None, + 'setting-up-the-equations-for-a-neural-network')]} end of tocinfo --> @@ -1056,21 +1074,20 @@

    Representing the wave function

    $$ - F_{rbm}(\mathbf{x},\mathbf{h}) = \frac{1}{Z} e^{-\frac{1}{T_0}E(\mathbf{x},\mathbf{h})}. + P_{rbm}(\mathbf{x},\mathbf{h}) = \frac{1}{Z} e^{-\frac{1}{T_0}E(\mathbf{x},\mathbf{h})}. $$

    To find the marginal distribution of \( \boldsymbol{x} \) we set:

    $$ - F_{rbm}(\mathbf{x}) =\frac{1}{Z}\sum_\mathbf{h} e^{-E(\mathbf{x}, \mathbf{h})}. + P_{rbm}(\mathbf{x}) =\frac{1}{Z}\sum_\mathbf{h} e^{-E(\mathbf{x}, \mathbf{h})}. $$

    Now this is what we use to represent the wave function, calling it a neural-network quantum state (NQS)

    $$ - \Psi (\mathbf{X}) = F_{rbm}(\mathbf{x}), + \vert\Psi (\mathbf{X})\vert^2 = P_{rbm}(\mathbf{x}). $$ -

    or we could square the wave function.











    Define the cost function

    @@ -1230,20 +1247,21 @@

    Basic Matrix Features

    or in terms of its column vectors \( \boldsymbol{a}_i \) as

    $$ - \mathbf{A} = + \boldsymbol{A} = \begin{bmatrix}\boldsymbol{a}_{0} & \boldsymbol{a}_{1} & \boldsymbol{a}_{2} & \dots & \dots & \boldsymbol{a}_{n-2} & \boldsymbol{a}_{n-1}\end{bmatrix}. $$ +

    We can think of a matrix as a diagram of in general \( n \) rowns and \( m \) columns. In the example here we have a square matrix.











    The inverse of a matrix

    -

    The inverse of a matrix (if it exists) is defined by

    +

    The inverse of a square matrix (if it exists) is defined by

    $$ -\mathbf{A}^{-1} \cdot \mathbf{A} = I, +\boldsymbol{A}^{-1} \cdot \boldsymbol{A} = I, $$

    where \( \boldsymbol{I} \) is the unit matrix.

    @@ -1292,15 +1310,15 @@

    Matrix Features

    Some Equivalent Statements

    -

    For an \( N\times N \) matrix \( \mathbf{A} \) the following properties are all equivalent

    +

    For an \( n\times n \) matrix \( \boldsymbol{A} \) the following properties are all equivalent

      -
    • If the inverse of \( \mathbf{A} \) exists, \( \mathbf{A} \) is nonsingular.
      • -
    • The equation \( \mathbf{Ax}=0 \) implies \( \mathbf{x}=0 \).
      • -
    • The rows of \( \mathbf{A} \) form a basis of \( R^N \).
      • -
    • The columns of \( \mathbf{A} \) form a basis of \( R^N \).
      • -
    • \( \mathbf{A} \) is a product of elementary matrices.
      • -
    • \( 0 \) is not eigenvalue of \( \mathbf{A} \).
      • +
    • If the inverse of \( \boldsymbol{A} \) exists, \( \boldsymbol{A} \) is nonsingular.
      • +
    • The equation \( \boldsymbol{Ax}=0 \) implies \( \boldsymbol{x}=0 \).
      • +
    • The rows of \( \boldsymbol{A} \) form a basis of \( R^N \).
      • +
    • The columns of \( \boldsymbol{A} \) form a basis of \( R^N \).
      • +
    • \( \boldsymbol{A} \) is a product of elementary matrices.
      • +
    • \( 0 \) is not an eigenvalue of \( \boldsymbol{A} \).
    @@ -1311,13 +1329,13 @@

    Important Mathematical Operations The basic matrix operations that we will deal with are addition and subtraction

    $$ -\mathbf{A}= \mathbf{B}\pm\mathbf{C} \Longrightarrow a_{ij} = b_{ij}\pm c_{ij}, +\boldsymbol{A}= \boldsymbol{B}\pm\boldsymbol{C} \Longrightarrow a_{ij} = b_{ij}\pm c_{ij}, $$

    and scalar-matrix multiplication

    $$ -\mathbf{A}= \gamma\mathbf{B} \Longrightarrow a_{ij} = \gamma b_{ij}. +\boldsymbol{A}= \gamma\boldsymbol{B} \Longrightarrow a_{ij} = \gamma b_{ij}. $$ @@ -1326,19 +1344,19 @@

    Vector-matrix and Matrix

    We have also vector-matrix multiplications

    $$ -\mathbf{y}=\mathbf{Ax} \Longrightarrow y_{i} = \sum_{j=1}^{n} a_{ij}x_j, +\boldsymbol{y}=\boldsymbol{Ax} \Longrightarrow y_{i} = \sum_{j=1}^{n} a_{ij}x_j, $$

    and matrix-matrix multiplications

    $$ -\mathbf{A}=\mathbf{BC} \Longrightarrow a_{ij} = \sum_{k=1}^{n} b_{ik}c_{kj}, +\boldsymbol{A}=\boldsymbol{BC} \Longrightarrow a_{ij} = \sum_{k=1}^{n} b_{ik}c_{kj}, $$

    and transpositions of a matrix

    $$ -\mathbf{A}=\mathbf{B}^T \Longrightarrow a_{ij} = b_{ji}. +\boldsymbol{A}=\boldsymbol{B}^T \Longrightarrow a_{ij} = b_{ji}. $$ @@ -1348,13 +1366,13 @@

    Important Mathematical Operations Similarly, important vector operations that we will deal with are addition and subtraction

    $$ -\mathbf{x}= \mathbf{y}\pm\mathbf{z} \Longrightarrow x_{i} = y_{i}\pm z_{i}, +\boldsymbol{x}= \boldsymbol{y}\pm\boldsymbol{z} \Longrightarrow x_{i} = y_{i}\pm z_{i}, $$

    scalar-vector multiplication

    $$ -\mathbf{x}= \gamma\mathbf{y} \Longrightarrow x_{i} = \gamma y_{i}, +\boldsymbol{x}= \gamma\boldsymbol{y} \Longrightarrow x_{i} = \gamma y_{i}, $$ @@ -1362,32 +1380,127 @@

    Important Mathematical Operations Other important mathematical operations

    and vector-vector multiplication (called Hadamard multiplication)

    $$ -\mathbf{x}=\mathbf{yz} \Longrightarrow x_{i} = y_{i}z_i. +\boldsymbol{x}=\boldsymbol{yz} \Longrightarrow x_{i} = y_{i}z_i. $$

    Finally, as already metnioned, the inner or so-called dot product resulting in a constant

    $$ -x=\mathbf{y}^T\mathbf{z} \Longrightarrow x = \sum_{j=1}^{n} y_{j}z_{j}, +x=\boldsymbol{y}^T\boldsymbol{z} \Longrightarrow x = \sum_{j=1}^{n} y_{j}z_{j}, $$

    and the outer product, which yields a matrix,

    $$ -\mathbf{A}= \mathbf{yz}^T \Longrightarrow a_{ij} = y_{i}z_{j}, +\boldsymbol{A}= \boldsymbol{y}\boldsymbol{z}^T \Longrightarrow a_{ij} = y_{i}z_{j}, $$ +









    +

    Further mathematical notations

    +
      +
    1. For all/any \( \forall \)
      2. +
    2. Implies \( \implies \)
      3. +
    3. Equivalent \( \equiv \)
      4. +
    4. Real variable \( \mathbb{R} \)
      5. +
    5. Integer variable \( \mathbb{I} \)
      6. +
    6. Complex variable \( \mathbb{C} \)
      7. +










    Setting up the basic equations for neural networks

    -

    Neural networks, its so-called feed-forward form, where each +

    Neural networks, in its so-called feed-forward form, where each iterations contains a feed-forward stage and a back-propgagation stage, consist of series of affine matrix-matrix and matrix-vector multiplications. The unknown parameters (the so-called biases and -weights which deternine the architecture of a neural network), are uptaded iteratively +weights which deternine the architecture of a neural network), are +uptaded iteratively using the so-called back-propagation algorithm. +This algorithm corresponds to the so-called reverse mode of the +automatic differentation algorithm. These algorithms will be discussed +in more detail below.

    +

    We start however first with the definitions of the various variables which make up a neural network.

    + +









    +

    Overarching view of a neural network

    + +

    The architecture of a neural network defines our model. This model +aims at describing some function \( f(\boldsymbol{x} \) which aims at describing +some final result (outputs or tagrget values) given a specific inpput +\( \boldsymbol{x} \). Note that here \( \boldsymbol{y} \) and \( \boldsymbol{x} \) are not limited to be +vectors. +

    + +

    The architecture consists of

    +
      +
    1. An input and an output layer where the input layer is defined by the inputs \( \boldsymbol{x} \). The output layer produces the model ouput \( \boldsymbol{\tilde{y}} \) which is compared with the target value \( \boldsymbol{y} \)
      2. +
    2. A given number of hidden layers and neurons/nodes/units for each layer (this may vary)
      3. +
    3. A given activation function \( \sigma(\boldsymbol{z}) \) with arguments \( \boldsymbol{z} \) to be defined below. The activation functions may differ from layer to layer.
      4. +
    4. The last layer, normally called output layer has normally an activation function tailored to the specific problem
      5. +
    5. Finally we define a so-called cost or loss function which is used to gauge the quality of our model.
      6. +
    +









    +

    The optimzation problem

    + +

    The cost function is a function of the unknown parameters +\( \boldsymbol{\Theta} \) where the latter is a container for all possible +parameters needed to define a neural network +

    + +

    If we are dealing with a regression task a typical cost/loss function +is the mean squared error +

    +$$ +C(\boldsymbol{\Theta})=\frac{1}{n}\left\{\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\theta}\right)^T\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\theta}\right)\right\}. +$$ + +

    This function represents one of many possible ways to define +the so-called cost function. +

    + +

    For neural networks the parameters +\( \boldsymbol{\Theta} \) are given by the so-called weights and biases (to be +defined below). +

    + +

    The weights are given by matrix elements \( w_{ij}^{(l)} \) where the +superscript indicates the layer number. The biases are typically given +by vector elements representing each single node of a given layer, +that is \( b_j^{(l)} \). +

    + +









    +

    Other ingredients of a neural network

    + +

    Having defined the architecture of a neural network, the optimization +of the cost function with respect to the parameters \( \boldsymbol{\Theta} \), +involves the calculations of gradients and their optimization. The +gradients represent the derivatives of a multidimensional object and +are often approximated by various gradient methods, including +

    +
      +
    1. various quasi-Newton methods,
      2. +
    2. plain gradient descent (GD) with a constant learning rate \( \eta \),
      3. +
    3. GD with momentum and other approximations to the learning rates such as
      4. +
        +
      • Adapative gradient (ADAgrad)
        • +
      • Root mean-square propagation (RMSprop)
        • +
      • Adaptive gradient with momentum (ADAM) and many other
        • +
      +
    4. Stochastic gradient descent and various families of learning rate approximations
      5. +
    +









    +

    Other parameters

    + +

    In addition to the above, there are often additional hyperparamaters +which are included in the setup of a neural network. These will be +discussed below. +

    + +









    +

    Setting up the equations for a neural network

    +
    © 1999-2024, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license diff --git a/doc/pub/week1/ipynb/ipynb-week1-src.tar.gz b/doc/pub/week1/ipynb/ipynb-week1-src.tar.gz index 6d9e3c8643bc97064d57d6d21a51794abdaca0e1..eed536739f21c058eb40996ee13db9025ecc6ffc 100644 GIT binary patch delta 139 zcmWN_OBR6u06-Q(98|1I3b$>pPK-$ug zuJojtP$KEeK!y^_NX9aeM5Z#6R5F>%LYA_UwQOW7JK4)Yaw+5}Cpk+gm0aZd<@SsJ DP?8{| delta 139 zcmWN_xdFid06ljJ#D--A}VFVkgQc3ZeSHIpt-5`HPt?&JT1k#p{ zbfqWFgc3<#1~QabMlzO(Br=tmq>{;87P6F;tYsrx*~wlGl1m{+ImuZ{spKNp54T_Z E2Wws++5i9m diff --git a/doc/pub/week1/ipynb/week1.ipynb b/doc/pub/week1/ipynb/week1.ipynb index b25ab48f..3f44604a 100644 --- a/doc/pub/week1/ipynb/week1.ipynb +++ b/doc/pub/week1/ipynb/week1.ipynb @@ -2,7 +2,7 @@ "cells": [ { "cell_type": "markdown", - "id": "5aa8d6d5", + "id": "9145ee16", "metadata": { "editable": true }, @@ -14,7 +14,7 @@ }, { "cell_type": "markdown", - "id": "3a8d40c8", + "id": "0fbf74a1", "metadata": { "editable": true }, @@ -27,7 +27,7 @@ }, { "cell_type": "markdown", - "id": "f5ad12d7", + "id": "785fd198", "metadata": { "editable": true }, @@ -43,7 +43,7 @@ }, { "cell_type": "markdown", - "id": "b2402a1d", + "id": "2aa90abe", "metadata": { "editable": true }, @@ -61,7 +61,7 @@ }, { "cell_type": "markdown", - "id": "5e3bdce2", + "id": "ef56854f", "metadata": { "editable": true }, @@ -97,7 +97,7 @@ }, { "cell_type": "markdown", - "id": "aad78991", + "id": "31a78df5", "metadata": { "editable": true }, @@ -113,7 +113,7 @@ }, { "cell_type": "markdown", - "id": "d799fa9c", + "id": "d5f14145", "metadata": { "editable": true }, @@ -136,7 +136,7 @@ }, { "cell_type": "markdown", - "id": "954ae34a", + "id": "a70ef5bf", "metadata": { "editable": true }, @@ -152,7 +152,7 @@ }, { "cell_type": "markdown", - "id": "6cc648e8", + "id": "76f6943d", "metadata": { "editable": true }, @@ -170,7 +170,7 @@ }, { "cell_type": "markdown", - "id": "055f2fa7", + "id": "76df72b5", "metadata": { "editable": true }, @@ -186,7 +186,7 @@ }, { "cell_type": "markdown", - "id": "52ec2841", + "id": "e62b5395", "metadata": { "editable": true }, @@ -206,7 +206,7 @@ }, { "cell_type": "markdown", - "id": "1c77c6a1", + "id": "4387a337", "metadata": { "editable": true }, @@ -225,7 +225,7 @@ }, { "cell_type": "markdown", - "id": "d1f50242", + "id": "abe021f2", "metadata": { "editable": true }, @@ -243,7 +243,7 @@ }, { "cell_type": "markdown", - "id": "41956cae", + "id": "2662b56c", "metadata": { "editable": true }, @@ -265,7 +265,7 @@ }, { "cell_type": "markdown", - "id": "0b9a6346", + "id": "f367dae7", "metadata": { "editable": true }, @@ -288,7 +288,7 @@ }, { "cell_type": "markdown", - "id": "bb9af663", + "id": "96bcbd8f", "metadata": { "editable": true }, @@ -304,7 +304,7 @@ }, { "cell_type": "markdown", - "id": "02151c67", + "id": "f5805026", "metadata": { "editable": true }, @@ -330,7 +330,7 @@ }, { "cell_type": "markdown", - "id": "2d75b1b2", + "id": "2c93fcdc", "metadata": { "editable": true }, @@ -346,7 +346,7 @@ }, { "cell_type": "markdown", - "id": "e4e8e6b1", + "id": "616d4b20", "metadata": { "editable": true }, @@ -362,7 +362,7 @@ }, { "cell_type": "markdown", - "id": "6f223101", + "id": "1c84480f", "metadata": { "editable": true }, @@ -382,7 +382,7 @@ }, { "cell_type": "markdown", - "id": "433a86aa", + "id": "b0c2fb72", "metadata": { "editable": true }, @@ -398,7 +398,7 @@ }, { "cell_type": "markdown", - "id": "21ec1bda", + "id": "704975d9", "metadata": { "editable": true }, @@ -416,7 +416,7 @@ }, { "cell_type": "markdown", - "id": "782b2cde", + "id": "22fd30f6", "metadata": { "editable": true }, @@ -432,7 +432,7 @@ }, { "cell_type": "markdown", - "id": "f42d8776", + "id": "027e29f4", "metadata": { "editable": true }, @@ -444,7 +444,7 @@ }, { "cell_type": "markdown", - "id": "beb99998", + "id": "420a0b98", "metadata": { "editable": true }, @@ -462,7 +462,7 @@ }, { "cell_type": "markdown", - "id": "2cc82ec5", + "id": "eb9afd0a", "metadata": { "editable": true }, @@ -474,7 +474,7 @@ }, { "cell_type": "markdown", - "id": "cfdf7b89", + "id": "72445622", "metadata": { "editable": true }, @@ -486,7 +486,7 @@ }, { "cell_type": "markdown", - "id": "41d84a80", + "id": "fdeb110e", "metadata": { "editable": true }, @@ -498,7 +498,7 @@ }, { "cell_type": "markdown", - "id": "26eca682", + "id": "f0c0d680", "metadata": { "editable": true }, @@ -508,7 +508,7 @@ }, { "cell_type": "markdown", - "id": "8778ef88", + "id": "9719bc2d", "metadata": { "editable": true }, @@ -520,7 +520,7 @@ }, { "cell_type": "markdown", - "id": "714a5aa0", + "id": "d0d636e1", "metadata": { "editable": true }, @@ -530,7 +530,7 @@ }, { "cell_type": "markdown", - "id": "e93c9d78", + "id": "f59fa7c3", "metadata": { "editable": true }, @@ -542,7 +542,7 @@ }, { "cell_type": "markdown", - "id": "fd3177fe", + "id": "6e00b8c9", "metadata": { "editable": true }, @@ -554,7 +554,7 @@ }, { "cell_type": "markdown", - "id": "1810b784", + "id": "7d4bf912", "metadata": { "editable": true }, @@ -564,7 +564,7 @@ }, { "cell_type": "markdown", - "id": "7fea88ef", + "id": "fa61f09f", "metadata": { "editable": true }, @@ -576,7 +576,7 @@ }, { "cell_type": "markdown", - "id": "46917de0", + "id": "87aa0e75", "metadata": { "editable": true }, @@ -586,7 +586,7 @@ }, { "cell_type": "markdown", - "id": "90f792c9", + "id": "ff4f9ed0", "metadata": { "editable": true }, @@ -598,7 +598,7 @@ }, { "cell_type": "markdown", - "id": "3d60ec55", + "id": "c7d5fb4e", "metadata": { "editable": true }, @@ -610,7 +610,7 @@ }, { "cell_type": "markdown", - "id": "6e7b847e", + "id": "14b7c00d", "metadata": { "editable": true }, @@ -620,7 +620,7 @@ }, { "cell_type": "markdown", - "id": "07381952", + "id": "a359efb4", "metadata": { "editable": true }, @@ -633,7 +633,7 @@ }, { "cell_type": "markdown", - "id": "dbb643bb", + "id": "89887fcb", "metadata": { "editable": true }, @@ -643,7 +643,7 @@ }, { "cell_type": "markdown", - "id": "f698bebe", + "id": "2db95a84", "metadata": { "editable": true }, @@ -655,7 +655,7 @@ }, { "cell_type": "markdown", - "id": "324d7c9b", + "id": "7347afc3", "metadata": { "editable": true }, @@ -670,7 +670,7 @@ }, { "cell_type": "markdown", - "id": "6e06a21b", + "id": "95aab3f4", "metadata": { "editable": true }, @@ -683,7 +683,7 @@ }, { "cell_type": "markdown", - "id": "d53a9426", + "id": "496aa7c0", "metadata": { "editable": true }, @@ -695,7 +695,7 @@ }, { "cell_type": "markdown", - "id": "479f4dc8", + "id": "ef9859a1", "metadata": { "editable": true }, @@ -707,7 +707,7 @@ }, { "cell_type": "markdown", - "id": "9646e7d3", + "id": "b75c5534", "metadata": { "editable": true }, @@ -719,7 +719,7 @@ }, { "cell_type": "markdown", - "id": "1130e6e3", + "id": "067715ec", "metadata": { "editable": true }, @@ -729,7 +729,7 @@ }, { "cell_type": "markdown", - "id": "adfab35a", + "id": "47e1d860", "metadata": { "editable": true }, @@ -742,7 +742,7 @@ }, { "cell_type": "markdown", - "id": "da5af7ce", + "id": "88d0c711", "metadata": { "editable": true }, @@ -753,7 +753,7 @@ }, { "cell_type": "markdown", - "id": "e1bda6fa", + "id": "dc47500f", "metadata": { "editable": true }, @@ -765,7 +765,7 @@ }, { "cell_type": "markdown", - "id": "12c30609", + "id": "8688a54d", "metadata": { "editable": true }, @@ -776,7 +776,7 @@ }, { "cell_type": "markdown", - "id": "3a4c7c01", + "id": "cf850d52", "metadata": { "editable": true }, @@ -790,7 +790,7 @@ }, { "cell_type": "markdown", - "id": "716f6c5e", + "id": "7179db13", "metadata": { "editable": true }, @@ -800,7 +800,7 @@ }, { "cell_type": "markdown", - "id": "864feb5e", + "id": "6039198e", "metadata": { "editable": true }, @@ -812,7 +812,7 @@ }, { "cell_type": "markdown", - "id": "2f9746bd", + "id": "210989b2", "metadata": { "editable": true }, @@ -822,7 +822,7 @@ }, { "cell_type": "markdown", - "id": "1ed15281", + "id": "caf2d377", "metadata": { "editable": true }, @@ -836,7 +836,7 @@ }, { "cell_type": "markdown", - "id": "01230a66", + "id": "4fb6fe8a", "metadata": { "editable": true }, @@ -848,7 +848,7 @@ }, { "cell_type": "markdown", - "id": "0ed7544f", + "id": "43cc3b4a", "metadata": { "editable": true }, @@ -859,7 +859,7 @@ }, { "cell_type": "markdown", - "id": "f33be476", + "id": "08a00311", "metadata": { "editable": true }, @@ -873,7 +873,7 @@ }, { "cell_type": "markdown", - "id": "89e071cf", + "id": "9afb4d16", "metadata": { "editable": true }, @@ -884,7 +884,7 @@ }, { "cell_type": "markdown", - "id": "1be402b3", + "id": "96f5044d", "metadata": { "editable": true }, @@ -896,7 +896,7 @@ }, { "cell_type": "markdown", - "id": "30998385", + "id": "3f032a1f", "metadata": { "editable": true }, @@ -906,7 +906,7 @@ }, { "cell_type": "markdown", - "id": "1eb6ab2e", + "id": "4b87a5f1", "metadata": { "editable": true }, @@ -918,7 +918,7 @@ }, { "cell_type": "markdown", - "id": "b0b683f5", + "id": "c84da29f", "metadata": { "editable": true }, @@ -928,7 +928,7 @@ }, { "cell_type": "markdown", - "id": "d2518e5d", + "id": "5f337e5b", "metadata": { "editable": true }, @@ -943,7 +943,7 @@ }, { "cell_type": "markdown", - "id": "2a22795c", + "id": "60aa417b", "metadata": { "editable": true }, @@ -955,7 +955,7 @@ }, { "cell_type": "markdown", - "id": "b0f590e8", + "id": "5a6cdaa9", "metadata": { "editable": true }, @@ -966,7 +966,7 @@ }, { "cell_type": "markdown", - "id": "fa7dcd5c", + "id": "4adc1cef", "metadata": { "editable": true }, @@ -978,7 +978,7 @@ }, { "cell_type": "markdown", - "id": "9e9b1f42", + "id": "9a2d54ed", "metadata": { "editable": true }, @@ -989,7 +989,7 @@ }, { "cell_type": "markdown", - "id": "00354a31", + "id": "a6fe7e41", "metadata": { "editable": true }, @@ -1001,7 +1001,7 @@ }, { "cell_type": "markdown", - "id": "9300c6d4", + "id": "6381a4cc", "metadata": { "editable": true }, @@ -1011,7 +1011,7 @@ }, { "cell_type": "markdown", - "id": "82210332", + "id": "cce8d8a4", "metadata": { "editable": true }, @@ -1023,7 +1023,7 @@ }, { "cell_type": "markdown", - "id": "6e3fb9bc", + "id": "ecfa7e7f", "metadata": { "editable": true }, @@ -1033,7 +1033,7 @@ }, { "cell_type": "markdown", - "id": "dd87b362", + "id": "dba47926", "metadata": { "editable": true }, @@ -1045,7 +1045,7 @@ }, { "cell_type": "markdown", - "id": "cbf9dea6", + "id": "46d9164e", "metadata": { "editable": true }, @@ -1055,7 +1055,7 @@ }, { "cell_type": "markdown", - "id": "1be869d4", + "id": "d75eafc5", "metadata": { "editable": true }, @@ -1072,7 +1072,7 @@ }, { "cell_type": "markdown", - "id": "0d7e9402", + "id": "5ac8e54a", "metadata": { "editable": true }, @@ -1088,7 +1088,7 @@ }, { "cell_type": "markdown", - "id": "4e04193b", + "id": "69f79239", "metadata": { "editable": true }, @@ -1104,7 +1104,7 @@ }, { "cell_type": "markdown", - "id": "ec153a0a", + "id": "1837d1e4", "metadata": { "editable": true }, @@ -1128,7 +1128,7 @@ }, { "cell_type": "markdown", - "id": "95b32165", + "id": "1d095fad", "metadata": { "editable": true }, @@ -1144,7 +1144,7 @@ }, { "cell_type": "markdown", - "id": "9b6aab50", + "id": "71238544", "metadata": { "editable": true }, @@ -1159,7 +1159,7 @@ }, { "cell_type": "markdown", - "id": "0b494c21", + "id": "dc675947", "metadata": { "editable": true }, @@ -1183,7 +1183,7 @@ }, { "cell_type": "markdown", - "id": "2f04d8a6", + "id": "5540c5d3", "metadata": { "editable": true }, @@ -1194,7 +1194,7 @@ }, { "cell_type": "markdown", - "id": "42992b55", + "id": "6f679b6a", "metadata": { "editable": true }, @@ -1206,7 +1206,7 @@ }, { "cell_type": "markdown", - "id": "c9cbeb9f", + "id": "d4131f17", "metadata": { "editable": true }, @@ -1216,7 +1216,7 @@ }, { "cell_type": "markdown", - "id": "e194bd37", + "id": "4a81072e", "metadata": { "editable": true }, @@ -1228,7 +1228,7 @@ }, { "cell_type": "markdown", - "id": "32c05ffc", + "id": "08427d74", "metadata": { "editable": true }, @@ -1238,7 +1238,7 @@ }, { "cell_type": "markdown", - "id": "286ce06b", + "id": "e42b46a4", "metadata": { "editable": true }, @@ -1255,7 +1255,7 @@ }, { "cell_type": "markdown", - "id": "eb092bd9", + "id": "fef4e721", "metadata": { "editable": true }, @@ -1271,7 +1271,7 @@ }, { "cell_type": "markdown", - "id": "31e20025", + "id": "91743971", "metadata": { "editable": true }, @@ -1283,7 +1283,7 @@ }, { "cell_type": "markdown", - "id": "75f46948", + "id": "f76256e7", "metadata": { "editable": true }, @@ -1293,7 +1293,7 @@ }, { "cell_type": "markdown", - "id": "6ec83924", + "id": "7c5cc1b8", "metadata": { "editable": true }, @@ -1307,7 +1307,7 @@ }, { "cell_type": "markdown", - "id": "1bf12a48", + "id": "110023c4", "metadata": { "editable": true }, @@ -1319,7 +1319,7 @@ }, { "cell_type": "markdown", - "id": "12b55322", + "id": "f94a0171", "metadata": { "editable": true }, @@ -1333,19 +1333,19 @@ }, { "cell_type": "markdown", - "id": "fc3825f9", + "id": "87720056", "metadata": { "editable": true }, "source": [ "$$\n", - "F_{rbm}(\\mathbf{x},\\mathbf{h}) = \\frac{1}{Z} e^{-\\frac{1}{T_0}E(\\mathbf{x},\\mathbf{h})}.\n", + "P_{rbm}(\\mathbf{x},\\mathbf{h}) = \\frac{1}{Z} e^{-\\frac{1}{T_0}E(\\mathbf{x},\\mathbf{h})}.\n", "$$" ] }, { "cell_type": "markdown", - "id": "907b6508", + "id": "b0437577", "metadata": { "editable": true }, @@ -1355,19 +1355,19 @@ }, { "cell_type": "markdown", - "id": "464e24c2", + "id": "2578c52d", "metadata": { "editable": true }, "source": [ "$$\n", - "F_{rbm}(\\mathbf{x}) =\\frac{1}{Z}\\sum_\\mathbf{h} e^{-E(\\mathbf{x}, \\mathbf{h})}.\n", + "P_{rbm}(\\mathbf{x}) =\\frac{1}{Z}\\sum_\\mathbf{h} e^{-E(\\mathbf{x}, \\mathbf{h})}.\n", "$$" ] }, { "cell_type": "markdown", - "id": "c9e7d52f", + "id": "32ba79ad", "metadata": { "editable": true }, @@ -1377,29 +1377,19 @@ }, { "cell_type": "markdown", - "id": "424ffd26", + "id": "5fbffb09", "metadata": { "editable": true }, "source": [ "$$\n", - "\\Psi (\\mathbf{X}) = F_{rbm}(\\mathbf{x}),\n", + "\\vert\\Psi (\\mathbf{X})\\vert^2 = P_{rbm}(\\mathbf{x}).\n", "$$" ] }, { "cell_type": "markdown", - "id": "05b8bc9e", - "metadata": { - "editable": true - }, - "source": [ - "or we could square the wave function." - ] - }, - { - "cell_type": "markdown", - "id": "29fe4c00", + "id": "47e799c4", "metadata": { "editable": true }, @@ -1418,7 +1408,7 @@ }, { "cell_type": "markdown", - "id": "56cdab5b", + "id": "c92a1da1", "metadata": { "editable": true }, @@ -1431,7 +1421,7 @@ }, { "cell_type": "markdown", - "id": "d2f9eeea", + "id": "801b4e49", "metadata": { "editable": true }, @@ -1441,7 +1431,7 @@ }, { "cell_type": "markdown", - "id": "3451f0a0", + "id": "c8566c22", "metadata": { "editable": true }, @@ -1453,7 +1443,7 @@ }, { "cell_type": "markdown", - "id": "5e3e1111", + "id": "f6ecf4c0", "metadata": { "editable": true }, @@ -1472,7 +1462,7 @@ }, { "cell_type": "markdown", - "id": "b3b71cb1", + "id": "fe4cc35d", "metadata": { "editable": true }, @@ -1494,7 +1484,7 @@ }, { "cell_type": "markdown", - "id": "54c5234f", + "id": "d2005a38", "metadata": { "editable": true }, @@ -1506,7 +1496,7 @@ }, { "cell_type": "markdown", - "id": "1a4c7361", + "id": "be359db9", "metadata": { "editable": true }, @@ -1518,7 +1508,7 @@ }, { "cell_type": "markdown", - "id": "7efb831f", + "id": "f837c54d", "metadata": { "editable": true }, @@ -1533,7 +1523,7 @@ }, { "cell_type": "markdown", - "id": "0afdfecf", + "id": "0d99c125", "metadata": { "editable": true }, @@ -1551,7 +1541,7 @@ }, { "cell_type": "markdown", - "id": "c425c8a4", + "id": "b88293fb", "metadata": { "editable": true }, @@ -1571,7 +1561,7 @@ }, { "cell_type": "markdown", - "id": "de6fa4c3", + "id": "4d574e8d", "metadata": { "editable": true }, @@ -1583,7 +1573,7 @@ }, { "cell_type": "markdown", - "id": "7f50df27", + "id": "7b10df3e", "metadata": { "editable": true }, @@ -1595,7 +1585,7 @@ }, { "cell_type": "markdown", - "id": "63815aa6", + "id": "eeeeb17f", "metadata": { "editable": true }, @@ -1605,7 +1595,7 @@ }, { "cell_type": "markdown", - "id": "1a5cf031", + "id": "ec2ee247", "metadata": { "editable": true }, @@ -1617,7 +1607,7 @@ }, { "cell_type": "markdown", - "id": "605c7eb5", + "id": "9a7373b1", "metadata": { "editable": true }, @@ -1627,7 +1617,7 @@ }, { "cell_type": "markdown", - "id": "706ae5f4", + "id": "f416e343", "metadata": { "editable": true }, @@ -1639,7 +1629,7 @@ }, { "cell_type": "markdown", - "id": "b90a47c6", + "id": "808e7560", "metadata": { "editable": true }, @@ -1649,7 +1639,7 @@ }, { "cell_type": "markdown", - "id": "f6e966aa", + "id": "9fbf3754", "metadata": { "editable": true }, @@ -1661,7 +1651,7 @@ }, { "cell_type": "markdown", - "id": "3d9ff6ba", + "id": "d34af01a", "metadata": { "editable": true }, @@ -1675,7 +1665,7 @@ }, { "cell_type": "markdown", - "id": "3ca238a4", + "id": "68010815", "metadata": { "editable": true }, @@ -1694,7 +1684,7 @@ }, { "cell_type": "markdown", - "id": "ce96d8e2", + "id": "2a094320", "metadata": { "editable": true }, @@ -1704,7 +1694,7 @@ }, { "cell_type": "markdown", - "id": "fac94317", + "id": "4d73dc30", "metadata": { "editable": true }, @@ -1716,7 +1706,7 @@ }, { "cell_type": "markdown", - "id": "c1b792d7", + "id": "29c0b9ba", "metadata": { "editable": true }, @@ -1735,7 +1725,7 @@ }, { "cell_type": "markdown", - "id": "3c09dcfd", + "id": "2c3d7c83", "metadata": { "editable": true }, @@ -1745,43 +1735,53 @@ }, { "cell_type": "markdown", - "id": "58e15900", + "id": "0e81bc3f", "metadata": { "editable": true }, "source": [ "$$\n", - "\\mathbf{A} =\n", + "\\boldsymbol{A} =\n", "\\begin{bmatrix}\\boldsymbol{a}_{0} & \\boldsymbol{a}_{1} & \\boldsymbol{a}_{2} & \\dots & \\dots & \\boldsymbol{a}_{n-2} & \\boldsymbol{a}_{n-1}\\end{bmatrix}.\n", "$$" ] }, { "cell_type": "markdown", - "id": "b66aa185", + "id": "e492390f", + "metadata": { + "editable": true + }, + "source": [ + "We can think of a matrix as a diagram of in general $n$ rowns and $m$ columns. In the example here we have a square matrix." + ] + }, + { + "cell_type": "markdown", + "id": "57dbfef2", "metadata": { "editable": true }, "source": [ "## The inverse of a matrix\n", - "The inverse of a matrix (if it exists) is defined by" + "The inverse of a square matrix (if it exists) is defined by" ] }, { "cell_type": "markdown", - "id": "9d280ab6", + "id": "9aa48b58", "metadata": { "editable": true }, "source": [ "$$\n", - "\\mathbf{A}^{-1} \\cdot \\mathbf{A} = I,\n", + "\\boldsymbol{A}^{-1} \\cdot \\boldsymbol{A} = I,\n", "$$" ] }, { "cell_type": "markdown", - "id": "8d9b1527", + "id": "5a5c05ed", "metadata": { "editable": true }, @@ -1791,7 +1791,7 @@ }, { "cell_type": "markdown", - "id": "16ecc48e", + "id": "4c953ae3", "metadata": { "editable": true }, @@ -1816,7 +1816,7 @@ }, { "cell_type": "markdown", - "id": "7651923b", + "id": "16c6f535", "metadata": { "editable": true }, @@ -1844,7 +1844,7 @@ }, { "cell_type": "markdown", - "id": "a74e540b", + "id": "7de44b1b", "metadata": { "editable": true }, @@ -1853,24 +1853,24 @@ "\n", "**Some Equivalent Statements.**\n", "\n", - "For an $N\\times N$ matrix $\\mathbf{A}$ the following properties are all equivalent\n", + "For an $n\\times n$ matrix $\\boldsymbol{A}$ the following properties are all equivalent\n", "\n", - " * If the inverse of $\\mathbf{A}$ exists, $\\mathbf{A}$ is nonsingular.\n", + " * If the inverse of $\\boldsymbol{A}$ exists, $\\boldsymbol{A}$ is nonsingular.\n", "\n", - " * The equation $\\mathbf{Ax}=0$ implies $\\mathbf{x}=0$.\n", + " * The equation $\\boldsymbol{Ax}=0$ implies $\\boldsymbol{x}=0$.\n", "\n", - " * The rows of $\\mathbf{A}$ form a basis of $R^N$.\n", + " * The rows of $\\boldsymbol{A}$ form a basis of $R^N$.\n", "\n", - " * The columns of $\\mathbf{A}$ form a basis of $R^N$.\n", + " * The columns of $\\boldsymbol{A}$ form a basis of $R^N$.\n", "\n", - " * $\\mathbf{A}$ is a product of elementary matrices.\n", + " * $\\boldsymbol{A}$ is a product of elementary matrices.\n", "\n", - " * $0$ is not eigenvalue of $\\mathbf{A}$." + " * $0$ is not an eigenvalue of $\\boldsymbol{A}$." ] }, { "cell_type": "markdown", - "id": "2f2e0757", + "id": "53b324ff", "metadata": { "editable": true }, @@ -1882,19 +1882,19 @@ }, { "cell_type": "markdown", - "id": "b7cc9ac8", + "id": "cfac91fe", "metadata": { "editable": true }, "source": [ "$$\n", - "\\mathbf{A}= \\mathbf{B}\\pm\\mathbf{C} \\Longrightarrow a_{ij} = b_{ij}\\pm c_{ij},\n", + "\\boldsymbol{A}= \\boldsymbol{B}\\pm\\boldsymbol{C} \\Longrightarrow a_{ij} = b_{ij}\\pm c_{ij},\n", "$$" ] }, { "cell_type": "markdown", - "id": "7c7a8dd2", + "id": "0c80d567", "metadata": { "editable": true }, @@ -1904,19 +1904,19 @@ }, { "cell_type": "markdown", - "id": "b9797790", + "id": "09b53bb3", "metadata": { "editable": true }, "source": [ "$$\n", - "\\mathbf{A}= \\gamma\\mathbf{B} \\Longrightarrow a_{ij} = \\gamma b_{ij}.\n", + "\\boldsymbol{A}= \\gamma\\boldsymbol{B} \\Longrightarrow a_{ij} = \\gamma b_{ij}.\n", "$$" ] }, { "cell_type": "markdown", - "id": "acd30cd2", + "id": "ebf9dd3c", "metadata": { "editable": true }, @@ -1928,19 +1928,19 @@ }, { "cell_type": "markdown", - "id": "5076e716", + "id": "770e2e6d", "metadata": { "editable": true }, "source": [ "$$\n", - "\\mathbf{y}=\\mathbf{Ax} \\Longrightarrow y_{i} = \\sum_{j=1}^{n} a_{ij}x_j,\n", + "\\boldsymbol{y}=\\boldsymbol{Ax} \\Longrightarrow y_{i} = \\sum_{j=1}^{n} a_{ij}x_j,\n", "$$" ] }, { "cell_type": "markdown", - "id": "9a50fb95", + "id": "30147ce1", "metadata": { "editable": true }, @@ -1950,19 +1950,19 @@ }, { "cell_type": "markdown", - "id": "4540905d", + "id": "0a207a12", "metadata": { "editable": true }, "source": [ "$$\n", - "\\mathbf{A}=\\mathbf{BC} \\Longrightarrow a_{ij} = \\sum_{k=1}^{n} b_{ik}c_{kj},\n", + "\\boldsymbol{A}=\\boldsymbol{BC} \\Longrightarrow a_{ij} = \\sum_{k=1}^{n} b_{ik}c_{kj},\n", "$$" ] }, { "cell_type": "markdown", - "id": "f88491aa", + "id": "5170d5b8", "metadata": { "editable": true }, @@ -1972,19 +1972,19 @@ }, { "cell_type": "markdown", - "id": "1ab50f86", + "id": "f801e7e9", "metadata": { "editable": true }, "source": [ "$$\n", - "\\mathbf{A}=\\mathbf{B}^T \\Longrightarrow a_{ij} = b_{ji}.\n", + "\\boldsymbol{A}=\\boldsymbol{B}^T \\Longrightarrow a_{ij} = b_{ji}.\n", "$$" ] }, { "cell_type": "markdown", - "id": "f9738980", + "id": "8c7e123c", "metadata": { "editable": true }, @@ -1996,19 +1996,19 @@ }, { "cell_type": "markdown", - "id": "2d19356a", + "id": "cb3f659b", "metadata": { "editable": true }, "source": [ "$$\n", - "\\mathbf{x}= \\mathbf{y}\\pm\\mathbf{z} \\Longrightarrow x_{i} = y_{i}\\pm z_{i},\n", + "\\boldsymbol{x}= \\boldsymbol{y}\\pm\\boldsymbol{z} \\Longrightarrow x_{i} = y_{i}\\pm z_{i},\n", "$$" ] }, { "cell_type": "markdown", - "id": "1c89fbf8", + "id": "60665aa6", "metadata": { "editable": true }, @@ -2018,19 +2018,19 @@ }, { "cell_type": "markdown", - "id": "f80368a0", + "id": "ab0f64c2", "metadata": { "editable": true }, "source": [ "$$\n", - "\\mathbf{x}= \\gamma\\mathbf{y} \\Longrightarrow x_{i} = \\gamma y_{i},\n", + "\\boldsymbol{x}= \\gamma\\boldsymbol{y} \\Longrightarrow x_{i} = \\gamma y_{i},\n", "$$" ] }, { "cell_type": "markdown", - "id": "9f68e0e4", + "id": "5fcc47f2", "metadata": { "editable": true }, @@ -2041,19 +2041,19 @@ }, { "cell_type": "markdown", - "id": "1f00c1c1", + "id": "eb67ed47", "metadata": { "editable": true }, "source": [ "$$\n", - "\\mathbf{x}=\\mathbf{yz} \\Longrightarrow x_{i} = y_{i}z_i.\n", + "\\boldsymbol{x}=\\boldsymbol{yz} \\Longrightarrow x_{i} = y_{i}z_i.\n", "$$" ] }, { "cell_type": "markdown", - "id": "7f2c1d5f", + "id": "8b87a1dd", "metadata": { "editable": true }, @@ -2063,19 +2063,19 @@ }, { "cell_type": "markdown", - "id": "da3e8d0a", + "id": "da7536cc", "metadata": { "editable": true }, "source": [ "$$\n", - "x=\\mathbf{y}^T\\mathbf{z} \\Longrightarrow x = \\sum_{j=1}^{n} y_{j}z_{j},\n", + "x=\\boldsymbol{y}^T\\boldsymbol{z} \\Longrightarrow x = \\sum_{j=1}^{n} y_{j}z_{j},\n", "$$" ] }, { "cell_type": "markdown", - "id": "131c91a3", + "id": "a09f7d10", "metadata": { "editable": true }, @@ -2085,30 +2085,186 @@ }, { "cell_type": "markdown", - "id": "4012fecb", + "id": "f00bfa90", "metadata": { "editable": true }, "source": [ "$$\n", - "\\mathbf{A}= \\mathbf{yz}^T \\Longrightarrow a_{ij} = y_{i}z_{j},\n", + "\\boldsymbol{A}= \\boldsymbol{y}\\boldsymbol{z}^T \\Longrightarrow a_{ij} = y_{i}z_{j},\n", "$$" ] }, { "cell_type": "markdown", - "id": "c1ffc4ae", + "id": "3b05f4a7", + "metadata": { + "editable": true + }, + "source": [ + "## Further mathematical notations\n", + "1. For all/any $\\forall$\n", + "\n", + "2. Implies $\\implies$\n", + "\n", + "3. Equivalent $\\equiv$\n", + "\n", + "4. Real variable $\\mathbb{R}$\n", + "\n", + "5. Integer variable $\\mathbb{I}$\n", + "\n", + "6. Complex variable $\\mathbb{C}$" + ] + }, + { + "cell_type": "markdown", + "id": "d7bd89f7", "metadata": { "editable": true }, "source": [ "## Setting up the basic equations for neural networks\n", "\n", - "Neural networks, its so-called feed-forward form, where each\n", + "Neural networks, in its so-called feed-forward form, where each\n", "iterations contains a feed-forward stage and a back-propgagation\n", "stage, consist of series of affine matrix-matrix and matrix-vector\n", "multiplications. The unknown parameters (the so-called biases and\n", - "weights which deternine the architecture of a neural network), are uptaded iteratively" + "weights which deternine the architecture of a neural network), are\n", + "uptaded iteratively using the so-called back-propagation algorithm.\n", + "This algorithm corresponds to the so-called reverse mode of the\n", + "automatic differentation algorithm. These algorithms will be discussed\n", + "in more detail below.\n", + "\n", + "We start however first with the definitions of the various variables which make up a neural network." + ] + }, + { + "cell_type": "markdown", + "id": "5866f84b", + "metadata": { + "editable": true + }, + "source": [ + "## Overarching view of a neural network\n", + "\n", + "The architecture of a neural network defines our model. This model\n", + "aims at describing some function $f(\\boldsymbol{x}$ which aims at describing\n", + "some final result (outputs or tagrget values) given a specific inpput\n", + "$\\boldsymbol{x}$. Note that here $\\boldsymbol{y}$ and $\\boldsymbol{x}$ are not limited to be\n", + "vectors.\n", + "\n", + "The architecture consists of\n", + "1. An input and an output layer where the input layer is defined by the inputs $\\boldsymbol{x}$. The output layer produces the model ouput $\\boldsymbol{\\tilde{y}}$ which is compared with the target value $\\boldsymbol{y}$\n", + "\n", + "2. A given number of hidden layers and neurons/nodes/units for each layer (this may vary)\n", + "\n", + "3. A given activation function $\\sigma(\\boldsymbol{z})$ with arguments $\\boldsymbol{z}$ to be defined below. The activation functions may differ from layer to layer.\n", + "\n", + "4. The last layer, normally called **output** layer has normally an activation function tailored to the specific problem\n", + "\n", + "5. Finally we define a so-called cost or loss function which is used to gauge the quality of our model." + ] + }, + { + "cell_type": "markdown", + "id": "523a5a8d", + "metadata": { + "editable": true + }, + "source": [ + "## The optimzation problem\n", + "\n", + "The cost function is a function of the unknown parameters\n", + "$\\boldsymbol{\\Theta}$ where the latter is a container for all possible\n", + "parameters needed to define a neural network\n", + "\n", + "If we are dealing with a regression task a typical cost/loss function\n", + "is the mean squared error" + ] + }, + { + "cell_type": "markdown", + "id": "2e5ead53", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "C(\\boldsymbol{\\Theta})=\\frac{1}{n}\\left\\{\\left(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\theta}\\right)^T\\left(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\theta}\\right)\\right\\}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "33bf6f95", + "metadata": { + "editable": true + }, + "source": [ + "This function represents one of many possible ways to define\n", + "the so-called cost function.\n", + "\n", + "For neural networks the parameters\n", + "$\\boldsymbol{\\Theta}$ are given by the so-called weights and biases (to be\n", + "defined below).\n", + "\n", + "The weights are given by matrix elements $w_{ij}^{(l)}$ where the\n", + "superscript indicates the layer number. The biases are typically given\n", + "by vector elements representing each single node of a given layer,\n", + "that is $b_j^{(l)}$." + ] + }, + { + "cell_type": "markdown", + "id": "f2ec1965", + "metadata": { + "editable": true + }, + "source": [ + "## Other ingredients of a neural network\n", + "\n", + "Having defined the architecture of a neural network, the optimization\n", + "of the cost function with respect to the parameters $\\boldsymbol{\\Theta}$,\n", + "involves the calculations of gradients and their optimization. The\n", + "gradients represent the derivatives of a multidimensional object and\n", + "are often approximated by various gradient methods, including\n", + "1. various quasi-Newton methods,\n", + "\n", + "2. plain gradient descent (GD) with a constant learning rate $\\eta$,\n", + "\n", + "3. GD with momentum and other approximations to the learning rates such as\n", + "\n", + " * Adapative gradient (ADAgrad)\n", + "\n", + " * Root mean-square propagation (RMSprop)\n", + "\n", + " * Adaptive gradient with momentum (ADAM) and many other\n", + "\n", + "4. Stochastic gradient descent and various families of learning rate approximations" + ] + }, + { + "cell_type": "markdown", + "id": "83b395d4", + "metadata": { + "editable": true + }, + "source": [ + "## Other parameters\n", + "\n", + "In addition to the above, there are often additional hyperparamaters\n", + "which are included in the setup of a neural network. These will be\n", + "discussed below." + ] + }, + { + "cell_type": "markdown", + "id": "6508d00a", + "metadata": { + "editable": true + }, + "source": [ + "## Setting up the equations for a neural network" ] } ], diff --git a/doc/pub/week1/pdf/week1.pdf b/doc/pub/week1/pdf/week1.pdf index c0eea9ea4779060bf3887d5906096a7684d9f9ac..b39b9c3c371529e6cb849ba3311b2701f1631a73 100644 GIT binary patch delta 94817 zcmZs>Q*Xqo{(au_rbucmUc4-(okyp&<7j}e+})ZI~z?q=k}nby2yqt zi=5D#RhT6O0nCe~BDwijuEO@tM0+TPA~vorYEuQ!KlkC{~Q3s!rN z%`DzxHxW2s4a4`DFmP?%VvQ7TYA9A6*4+hu;3CIO0Z4j>%e(3>OX6|o`~~KWbh$l} z60_kelf7@1nc3pR)gx8?K>4OhfbKZn%z;l?*R|~J4BX!xe6LRh?|Q4O`AvO_8h7ng z7gRikgqZ;?Jh@8^`XvTXw&nW^Ma5I?mP!(dg+`^1CIQ(Tsy5e>$wW~{+pKmm@f=f9S*0QzY1_+^=V(V4#kgv@%HTi2`_iR4bTG1_gA@uAV`xe*C2zbn zL%GpyN^eEwYyoTwm~oyQTz1Xcko_L_e+>{1aLKm)s!pXlHs(+&ZabyXo%~TzujsUK2rYV8OJ1(81dxje&%P{K-OHNY0qDz;2pQ7 zX$_UAUH0*l6!$?sw1YF;|GsBevKIysY zB;`*OoD_mSRV=`ayvPJKZSHv=HbjB*u9`_hY_yy-{{+JNys6+LHgm_u{@KI zAz={)eH$%0ypr{!2c)%So(l-0r9sDkNgXnSUcX#cv!-zgalY~+s1BE({s4T90S;VW z*&3X#M|J=XQ=zDp+Nr<2>qnj&J`oEBr|;j$KMQH*s@!9aAOY$emK5Y8yWIiYGP^h} z=@|W~69qHILG$Kt$?La=25_^?16kml^F=QKaw!9RXA-){$yC2W<0rl^1BI`*cP2zO zlNSnjL)IVQ&(ylT&jAEYwWy|UWLWs+9Z%X3(P@OALk9#4yOhm+S_sHDwZ>@)+CNZh z%tR$=!A_*;5$hO~{{_{5LH%FQw5?;%UM4}XvaqrIkBl7a%((sKLg+oE^+`q^MEkVn z7xb{sRN`L}_ZCsOa*6k`Rd~8v?k9;AjF&iVg{cKftc8H=VLM7+nYtnk&G%HW?jeOSt7XlgUF zqbTAQ-2KWhL~yJ2o#J+sq7Dj&U8xNV-JNGwL7p6+&_L0gf-*aHLCPFIstuiA zGseqVh7=0hWEEJRq}71_Zi6jK%?ql++=#DI_EEp=_HSop+gHm)3pz%r#K8uR0uf!- z)Mo&u=;%62y4PK~l(8wJxi-9r9NH&IRv`)hAsK@#Z&-v94#yg~$6XBFeJ4H`J3Mj0 z0N+=Lxj^KE?pzuep-b5~QLzOzawMgIoyDOlIeK24!kdD{B(R;H?a4GT@~3xvSY68M zf<8(?9=lPA+IVM-1zZ|% z%aM>gymc%WI`(LaSb`=>oFPh%?XJ%d5sZ|5wWONpQQ$+E*$`-TT$F!EUnHr?J8Jmj zaP^c#D)<o|j)Jdx zov^s!Qm51gURQ6-EX*vK_V%tG<58~>@qh-{pSn=7(?2+c+Yle6;aX4vFP&73*8wI_P7WjNQW zB|0xR8^*ikhp;gX93picp#mq^tv-2HUKY7>r@DHcd-tb&cD^u4em4|U4P(m$nhf_y zP+JXb-*~18GGks5F%hsBW4)tQpoM94bY&PvfU^@;l`wcss8FmXPWY;&ydHpBJ}vs9 zvm*7}*^sTkRT^Duyw#Z@&o)m^cvx=9GE2G2=<9;b&-g>thw%aWkTJzAJ=u-4sm4@` zjxvh)-M~jZLiVde@7oh!*y)05R%m8CAy-duSQj(=Y3!rY@HS%zSM1D3cq8#8#FfPn zejDiYntpfimiz7f&;$T!Dgra_PcyHgr2*q&PfPp;qXo=p2)Zwcpj>~^w-d!F z3ZM^#cjM+P+Ih&e4qfk*Y80G3xOzDOt0xv~rvL$~$kdAOzih zVe;|yPhWiTT-AeK*Ms8MbG11^Gq>=A2R6qodHw`8AM?ar7AvA)@K2=DM19I679Pc^ z+@Ht9NrjUS{R6e8LCp-sjB*(Yc2%rQr?B43bL9I4Ay4D-S?f~x)3Moh?;7Y8NXuY~ z23*dbMmG~xm2EHF#{p#QC1vf~YiOS@FO)wdlq#^qs7I>7 z<@piTEQ!b=YD1$VG~aP7T~)b%sM>3b_-6mLiL<#8z|s#UPy-x)Q}JN4LF(z&VJB|p zJG5(+E7g{q6}iqhsS{uP$~)r#mrVp&K1vfw;Y zjjvmrt~(Wwz6g}z)R$lGGru)av(ml1%WH0bP=7%LR(9u^28Fa&sw*%f7a7`c-;+$T z3FAAHu}J!R69GsN`8ZfSpfJ20CEEY)&d=4sk)!WZI6=f^9KKNQcJ3}T`@~dPTKi`x zh;q!#s56UgOE3h=r9Iv6Scu@TP$SQ$cqK$P>X^}J=`NR&I(+>6N>?yT;&aE#eBhKs z@X{!=WIb-g%9p%#O)E9K)MANYS+UP};V-9Ig%0)_m;&53Eo*oDt5DlVY_NZuP$#q# zIGX;Ofe)oIh0b30`E_+~&gYpjVA2J?$Vj_2mF=CjaruqOxU|5!1v#a{Qf=+K5{YBh zk9$E`@5CN5OLH35tGHnn#HutHy#8;xWvF&YdD>sqs;YZEnxG02^UyKjAuXYg*0o9` zKe3LicLs39`wyJXe7TK!X*&`nVr!-FhQ5`4>e<#@g

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