-
Figure 1: A regular 3-layer Neural Network.
-
@@ -446,7 +447,7 @@ MathJax.Hub.Config({
+
-What is the difference? CNN architectures make the explicit assumption that -the inputs are images, which allows us to encode certain properties -into the architecture. These then make the forward function more -efficient to implement and vastly reduce the amount of parameters in -the network. - -
-Here we provide only a superficial overview, for the more interested, we recommend highly the course -IN5400 – Machine Learning for Image Analysis -and the slides of CS231. - -
-Another good read is the article here https://arxiv.org/pdf/1603.07285.pdf. -
@@ -465,7 +451,7 @@ Another good read is the article here 60
-Neural networks are defined as affine transformations, that is -a vector is received as input and is multiplied with a matrix of so-called weights (our unknown paramters) to produce an -output (to which a bias vector is usually added before passing the result -through a nonlinear activation function). This is applicable to any type of input, be it an -image, a sound clip or an unordered collection of features: whatever their -dimensionality, their representation can always be flattened into a vector -before the transformation. +CNN architectures make the explicit assumption that +the inputs are images, which allows us to encode certain properties +into the architecture. These then make the forward function more +efficient to implement and vastly reduce the amount of parameters in +the network. + +
+Here we provide only a superficial overview, for the more interested, we recommend highly the course +IN5400 – Machine Learning for Image Analysis +and the slides of CS231. + +
+Another good read is the article here https://arxiv.org/pdf/1603.07285.pdf.
@@ -440,7 +447,7 @@ before the transformation.
-However, when we consider images, sound clips and many other similar kinds of data, these data have an intrinsic -structure. More formally, they share these important properties: - -
-A discrete convolution is a linear transformation that preserves this notion of -ordering. It is sparse (only a few input units contribute to a given output -unit) and reuses parameters (the same weights are applied to multiple locations -in the input). +Neural networks are defined as affine transformations, that is +a vector is received as input and is multiplied with a matrix of so-called weights (our unknown paramters) to produce an +output (to which a bias vector is usually added before passing the result +through a nonlinear activation function). This is applicable to any type of input, be it an +image, a sound clip or an unordered collection of features: whatever their +dimensionality, their representation can always be flattened into a vector +before the transformation.
@@ -454,7 +441,7 @@ in the input).
-As an example, consider -an image of size \( 32\times 32\times 3 \) (32 wide, 32 high, 3 color channels), so a -single fully-connected neuron in a first hidden layer of a regular -Neural Network would have \( 32\times 32\times 3 = 3072 \) weights. This amount still -seems manageable, but clearly this fully-connected structure does not -scale to larger images. For example, an image of more respectable -size, say \( 200\times 200\times 3 \), would lead to neurons that have -\( 200\times 200\times 3 = 120,000 \) weights. +However, when we consider images, sound clips and many other similar kinds of data, these data have an intrinsic +structure. More formally, they share these important properties: + +
-We could have -several such neurons, and the parameters would add up quickly! Clearly, -this full connectivity is wasteful and the huge number of parameters -would quickly lead to possible overfitting. - -
-
Figure 1: A regular 3-layer Neural Network.
@@ -454,7 +455,7 @@ would quickly lead to possible overfitting.
-Convolutional Neural Networks take advantage of the fact that the -input consists of images and they constrain the architecture in a more -sensible way. +As an example, consider +an image of size \( 32\times 32\times 3 \) (32 wide, 32 high, 3 color channels), so a +single fully-connected neuron in a first hidden layer of a regular +Neural Network would have \( 32\times 32\times 3 = 3072 \) weights. This amount still +seems manageable, but clearly this fully-connected structure does not +scale to larger images. For example, an image of more respectable +size, say \( 200\times 200\times 3 \), would lead to neurons that have +\( 200\times 200\times 3 = 120,000 \) weights.
-In particular, unlike a regular Neural Network, the -layers of a CNN have neurons arranged in 3 dimensions: width, -height, depth. (Note that the word depth here refers to the third -dimension of an activation volume, not to the depth of a full Neural -Network, which can refer to the total number of layers in a network.) - -
-To understand it better, the above example of an image -with an input volume of -activations has dimensions \( 32\times 32\times 3 \) (width, height, -depth respectively). - -
-The neurons in a layer will -only be connected to a small region of the layer before it, instead of -all of the neurons in a fully-connected manner. Moreover, the final -output layer could for this specific image have dimensions \( 1\times 1 \times 10 \), -because by the -end of the CNN architecture we will reduce the full image into a -single vector of class scores, arranged along the depth -dimension. +We could have +several such neurons, and the parameters would add up quickly! Clearly, +this full connectivity is wasteful and the huge number of parameters +would quickly lead to possible overfitting.
Figure 2: A CNN arranges its neurons in three dimensions (width, height, depth), as visualized in one of the layers. Every layer of a CNN transforms the 3D input volume to a 3D output volume of neuron activations. In this example, the red input layer holds the image, so its width and height would be the dimensions of the image, and the depth would be 3 (Red, Green, Blue channels).
Figure 1: A regular 3-layer Neural Network.
@@ -466,7 +455,7 @@ dimension.
- + -
-A simple CNN is a sequence of layers, and every layer of a CNN -transforms one volume of activations to another through a -differentiable function. We use three main types of layers to build -CNN architectures: Convolutional Layer, Pooling Layer, and -Fully-Connected Layer (exactly as seen in regular Neural Networks). We -will stack these layers to form a full CNN architecture. +Convolutional Neural Networks take advantage of the fact that the +input consists of images and they constrain the architecture in a more +sensible way.
-A simple CNN for image classification could have the architecture: +In particular, unlike a regular Neural Network, the +layers of a CNN have neurons arranged in 3 dimensions: width, +height, depth. (Note that the word depth here refers to the third +dimension of an activation volume, not to the depth of a full Neural +Network, which can refer to the total number of layers in a network.) -
+To understand it better, the above example of an image +with an input volume of +activations has dimensions \( 32\times 32\times 3 \) (width, height, +depth respectively). +
+The neurons in a layer will +only be connected to a small region of the layer before it, instead of +all of the neurons in a fully-connected manner. Moreover, the final +output layer could for this specific image have dimensions \( 1\times 1 \times 10 \), +because by the +end of the CNN architecture we will reduce the full image into a +single vector of class scores, arranged along the depth +dimension. + +
+
Figure 2: A CNN arranges its neurons in three dimensions (width, height, depth), as visualized in one of the layers. Every layer of a CNN transforms the 3D input volume to a 3D output volume of neuron activations. In this example, the red input layer holds the image, so its width and height would be the dimensions of the image, and the depth would be 3 (Red, Green, Blue channels).
- + -
-CNNs transform the original image layer by layer from the original -pixel values to the final class scores. +A simple CNN is a sequence of layers, and every layer of a CNN +transforms one volume of activations to another through a +differentiable function. We use three main types of layers to build +CNN architectures: Convolutional Layer, Pooling Layer, and +Fully-Connected Layer (exactly as seen in regular Neural Networks). We +will stack these layers to form a full CNN architecture.
-Observe that some layers contain -parameters and other don’t. In particular, the CNN layers perform -transformations that are a function of not only the activations in the -input volume, but also of the parameters (the weights and biases of -the neurons). On the other hand, the RELU/POOL layers will implement a -fixed function. The parameters in the CONV/FC layers will be trained -with gradient descent so that the class scores that the CNN computes -are consistent with the labels in the training set for each image. +A simple CNN for image classification could have the architecture: + +
-In summary: +CNNs transform the original image layer by layer from the original +pixel values to the final class scores. -
+Observe that some layers contain +parameters and other don’t. In particular, the CNN layers perform +transformations that are a function of not only the activations in the +input volume, but also of the parameters (the weights and biases of +the neurons). On the other hand, the RELU/POOL layers will implement a +fixed function. The parameters in the CONV/FC layers will be trained +with gradient descent so that the class scores that the CNN computes +are consistent with the labels in the training set for each image.
@@ -447,7 +446,7 @@ and the slides of 67
-The mathematics of CNNs is based on the mathematical operation of -convolution. In mathematics (in particular in functional analysis), -convolution is represented by matheematical operation (integration, -summation etc) on two function in order to produce a third function -that expresses how the shape of one gets modified by the other. -Convolution has a plethora of applications in a variety of disciplines, spanning from statistics to signal processing, computer vision, solutions of differential equations,linear algebra, engineering, and yes, machine learning. +In summary: -
-Mathematically, convolution is defined as follows (one-dimensional example): -Let us define a continuous function \( y(t) \) given by -$$ -y(t) = \int x(a) w(t-a) da, -$$ +
-The above integral is written in a more compact form as -$$ -y(t) = \left(x * w\right)(t). -$$ - -
-The discretized version reads -$$ -y(t) = \sum_{a=-\infty}^{a=\infty}x(a)w(t-a). -$$ - -Computing the inverse of the above convolution operations is known as deconvolution. - -
-How can we use this? And what does it mean? Let us study some familiar examples first. +For more material on convolutional networks, we strongly recommend +the course +IN5400 – Machine Learning for Image Analysis +and the slides of CS231 which is taught at Stanford University (consistently ranked as one of the top computer science programs in the world). Michael Nielsen's book is a must read, in particular chapter 6 which deals with CNNs.
@@ -465,7 +448,7 @@ How can we use this? And what does it mean? Let us study some familiar examples
-We have already met such an example in project 1 when we tried to set -up the design matrix for a two-dimensional function. This was an -example of polynomial multiplication. Let us recast such a problem in terms of the convolution operation. -Let us look a the following polynomials to second and third order, respectively: +The mathematics of CNNs is based on the mathematical operation of +convolution. In mathematics (in particular in functional analysis), +convolution is represented by matheematical operation (integration, +summation etc) on two function in order to produce a third function +that expresses how the shape of one gets modified by the other. +Convolution has a plethora of applications in a variety of disciplines, spanning from statistics to signal processing, computer vision, solutions of differential equations,linear algebra, engineering, and yes, machine learning. + +
+Mathematically, convolution is defined as follows (one-dimensional example): +Let us define a continuous function \( y(t) \) given by $$ -p(t) = \alpha_0+\alpha_1 t+\alpha_2 t^2, +y(t) = \int x(a) w(t-a) da, $$ -and +where \( x(a) \) represents a so-called input and \( w(t-a) \) is normally called the weight function or kernel. + +
+The above integral is written in a more compact form as $$ -s(t) = \beta_0+\beta_1 t+\beta_2 t^2+\beta_3 t^3. +y(t) = \left(x * w\right)(t). $$
-The polynomial multiplication gives us a new polynomial of degree \( 5 \) +The discretized version reads $$ -z(t) = \delta_0+\delta_1 t+\delta_2 t^2+\delta_3 t^3+\delta_4 t^4+\delta_5 t^5. +y(t) = \sum_{a=-\infty}^{a=\infty}x(a)w(t-a). $$ +Computing the inverse of the above convolution operations is known as deconvolution. + +
+How can we use this? And what does it mean? Let us study some familiar examples first. +
@@ -451,7 +466,7 @@ $$
-Computing polynomial products can be implemented efficiently if we rewrite the more brute force multiplications using convolution. -We note first that the new coefficients are given as - +We have already met such an example in project 1 when we tried to set +up the design matrix for a two-dimensional function. This was an +example of polynomial multiplication. Let us recast such a problem in terms of the convolution operation. +Let us look a the following polynomials to second and third order, respectively: $$ -\begin{split} -\delta_0=&\alpha_0\beta_0\\ -\delta_1=&\alpha_1\beta_0+\alpha_1\beta_0\\ -\delta_2=&\alpha_0\beta_2+\alpha_1\beta_1+\alpha_2\beta_0\\ -\delta_3=&\alpha_1\beta_2+\alpha_2\beta_1+\alpha_0\beta_3\\ -\delta_4=&\alpha_2\beta_2+\alpha_1\beta_3\\ -\delta_5=&\alpha_2\beta_3.\\ -\end{split} +p(t) = \alpha_0+\alpha_1 t+\alpha_2 t^2, +$$ + +and +$$ +s(t) = \beta_0+\beta_1 t+\beta_2 t^2+\beta_3 t^3. $$
-We note that \( \alpha_i=0 \) except for \( i\in \left{0,1,2\right} \) and \( \beta_i=0 \) except for \( i\in\left{0,1,2,3\right} \). - -
-We can then rewrite the coefficients \( \delta_j \) using a discrete convolution as +The polynomial multiplication gives us a new polynomial of degree \( 5 \) $$ -\delta_j = \sum_{i=-\infty}^{i=\infty}\alpha_i\beta_{j-i}=(\alpha * \beta)_j, +z(t) = \delta_0+\delta_1 t+\delta_2 t^2+\delta_3 t^3+\delta_4 t^4+\delta_5 t^5. $$ -or as a double sum with restriction \( l=i+j \) -$$ -\delta_l = \sum_{ij}\alpha_i\beta_{j}. -$$ - -
-Do you see a potential drawback with these equations? -
@@ -463,7 +452,7 @@ Do you see a potential drawback with these equations?
-Since we only have a finite number of \( \alpha \) and \( \beta \) values -which are non-zero, we can rewrite the above convolution expressions -as a matrix-vector multiplication +Computing polynomial products can be implemented efficiently if we rewrite the more brute force multiplications using convolution. +We note first that the new coefficients are given as $$ -\boldsymbol{\delta}=\begin{bmatrix}\alpha_0 & 0 & 0 & 0 \\ - \alpha_1 & \alpha_0 & 0 & 0 \\ - \alpha_2 & \alpha_1 & \alpha_0 & 0 \\ - 0 & \alpha_2 & \alpha_1 & \alpha_0 \\ - 0 & 0 & \alpha_2 & \alpha_1 \\ - 0 & 0 & 0 & \alpha_2 - \end{bmatrix}\begin{bmatrix} \beta_0 \\ \beta_1 \\ \beta_2 \\ \beta_3\end{bmatrix}. +\begin{split} +\delta_0=&\alpha_0\beta_0\\ +\delta_1=&\alpha_1\beta_0+\alpha_1\beta_0\\ +\delta_2=&\alpha_0\beta_2+\alpha_1\beta_1+\alpha_2\beta_0\\ +\delta_3=&\alpha_1\beta_2+\alpha_2\beta_1+\alpha_0\beta_3\\ +\delta_4=&\alpha_2\beta_2+\alpha_1\beta_3\\ +\delta_5=&\alpha_2\beta_3.\\ +\end{split} $$
-The process is commutative and we can easily see that we can rewrite the multiplication in terms of a matrix holding \( \beta \) and a vector holding \( \alpha \). -In this case we have +We note that \( \alpha_i=0 \) except for \( i\in \left{0,1,2\right} \) and \( \beta_i=0 \) except for \( i\in\left{0,1,2,3\right} \). + +
+We can then rewrite the coefficients \( \delta_j \) using a discrete convolution as $$ -\boldsymbol{\delta}=\begin{bmatrix}\beta_0 & 0 & 0 \\ - \beta_1 & \beta_0 & 0 \\ - \beta_2 & \beta_1 & \beta_0 \\ - \beta_3 & \beta_2 & \beta_1 \\ - 0 & \beta_3 & \beta_2 \\ - 0 & 0 & \beta_3 - \end{bmatrix}\begin{bmatrix} \alpha_0 \\ \alpha_1 \\ \alpha_2\end{bmatrix}. +\delta_j = \sum_{i=-\infty}^{i=\infty}\alpha_i\beta_{j-i}=(\alpha * \beta)_j, +$$ + +or as a double sum with restriction \( l=i+j \) +$$ +\delta_l = \sum_{ij}\alpha_i\beta_{j}. $$
-Note that the use of these matrices is for mathematical purposes only and not implementation purposes. -When implementing the above equation we do not encode (and allocate memory) the matrices explicitely. -We rather code the convolutions in the minimal memory footprint that they require. +Do you see a potential drawback with these equations?
@@ -464,7 +464,7 @@ We rather code the convolutions in the minimal memory footprint that they requir
-For problems with so-called harmonic oscillations, given by for example the following differential equation -$$ -m\frac{d^2x}{dt^2}+\eta\frac{dx}{dt}+x(t)=F(t), -$$ +Since we only have a finite number of \( \alpha \) and \( \beta \) values +which are non-zero, we can rewrite the above convolution expressions +as a matrix-vector multiplication -where \( F(t) \) is an applied external force acting on the system (often called a driving force), one can use the theory of Fourier transformations to find the solutions of this type of equations. +$$ +\boldsymbol{\delta}=\begin{bmatrix}\alpha_0 & 0 & 0 & 0 \\ + \alpha_1 & \alpha_0 & 0 & 0 \\ + \alpha_2 & \alpha_1 & \alpha_0 & 0 \\ + 0 & \alpha_2 & \alpha_1 & \alpha_0 \\ + 0 & 0 & \alpha_2 & \alpha_1 \\ + 0 & 0 & 0 & \alpha_2 + \end{bmatrix}\begin{bmatrix} \beta_0 \\ \beta_1 \\ \beta_2 \\ \beta_3\end{bmatrix}. +$$
-If one has several driving forces, \( F(t)=\sum_n F_n(t) \), one can find -the particular solution to each \( F_n \), \( x_{pn}(t) \), and the particular -solution for the entire driving force is then given by a series like +The process is commutative and we can easily see that we can rewrite the multiplication in terms of a matrix holding \( \beta \) and a vector holding \( \alpha \). +In this case we have +$$ +\boldsymbol{\delta}=\begin{bmatrix}\beta_0 & 0 & 0 \\ + \beta_1 & \beta_0 & 0 \\ + \beta_2 & \beta_1 & \beta_0 \\ + \beta_3 & \beta_2 & \beta_1 \\ + 0 & \beta_3 & \beta_2 \\ + 0 & 0 & \beta_3 + \end{bmatrix}\begin{bmatrix} \alpha_0 \\ \alpha_1 \\ \alpha_2\end{bmatrix}. +$$ -$$ -\begin{equation} -x_p(t)=\sum_nx_{pn}(t). -\tag{21} -\end{equation} -$$ +
+Note that the use of these matrices is for mathematical purposes only and not implementation purposes. +When implementing the above equation we do not encode (and allocate memory) the matrices explicitely. +We rather code the convolutions in the minimal memory footprint that they require.
@@ -451,7 +465,7 @@ $$
-This is known as the principle of superposition. It only applies when -the homogenous equation is linear. If there were an anharmonic term -such as \( x^3 \) in the homogenous equation, then when one summed various -solutions, \( x=(\sum_n x_n)^2 \), one would get cross -terms. Superposition is especially useful when \( F(t) \) can be written -as a sum of sinusoidal terms, because the solutions for each -sinusoidal (sine or cosine) term is analytic. - -
-Driving forces are often periodic, even when they are not -sinusoidal. Periodicity implies that for some time \( \tau \) - +For problems with so-called harmonic oscillations, given by for example the following differential equation $$ -\begin{eqnarray} -F(t+\tau)=F(t). -\end{eqnarray} +m\frac{d^2x}{dt^2}+\eta\frac{dx}{dt}+x(t)=F(t), $$ +where \( F(t) \) is an applied external force acting on the system (often called a driving force), one can use the theory of Fourier transformations to find the solutions of this type of equations. +
-One example of a non-sinusoidal periodic force is a square wave. Many -components in electric circuits are non-linear, e.g. diodes, which -makes many wave forms non-sinusoidal even when the circuits are being -driven by purely sinusoidal sources. +If one has several driving forces, \( F(t)=\sum_n F_n(t) \), one can find +the particular solution to each \( F_n \), \( x_{pn}(t) \), and the particular +solution for the entire driving force is then given by a series like + +$$ +\begin{equation} +x_p(t)=\sum_nx_{pn}(t). +\tag{21} +\end{equation} +$$
@@ -456,7 +452,7 @@ driven by purely sinusoidal sources.
-The code here shows a typical example of such a square wave generated using the functionality included in the scipy Python package. We have used a period of \( \tau=0.2 \). +This is known as the principle of superposition. It only applies when +the homogenous equation is linear. If there were an anharmonic term +such as \( x^3 \) in the homogenous equation, then when one summed various +solutions, \( x=(\sum_n x_n)^2 \), one would get cross +terms. Superposition is especially useful when \( F(t) \) can be written +as a sum of sinusoidal terms, because the solutions for each +sinusoidal (sine or cosine) term is analytic.
- - -
import numpy as np
-import math
-from scipy import signal
-import matplotlib.pyplot as plt
-
-# number of points
-n = 500
-# start and final times
-t0 = 0.0
-tn = 1.0
-# Period
-t = np.linspace(t0, tn, n, endpoint=False)
-SqrSignal = np.zeros(n)
-SqrSignal = 1.0+signal.square(2*np.pi*5*t)
-plt.plot(t, SqrSignal)
-plt.ylim(-0.5, 2.5)
-plt.show()
--For the sinusoidal example the -period is \( \tau=2\pi/\omega \). However, higher harmonics can also -satisfy the periodicity requirement. In general, any force that -satisfies the periodicity requirement can be expressed as a sum over -harmonics, +Driving forces are often periodic, even when they are not +sinusoidal. Periodicity implies that for some time \( \tau \) $$ -\begin{equation} -F(t)=\frac{f_0}{2}+\sum_{n>0} f_n\cos(2n\pi t/\tau)+g_n\sin(2n\pi t/\tau). -\tag{22} -\end{equation} +\begin{eqnarray} +F(t+\tau)=F(t). +\end{eqnarray} $$ +
+One example of a non-sinusoidal periodic force is a square wave. Many +components in electric circuits are non-linear, e.g. diodes, which +makes many wave forms non-sinusoidal even when the circuits are being +driven by purely sinusoidal sources. +
@@ -469,7 +457,7 @@ $$
-We can write down the answer for -\( x_{pn}(t) \), by substituting \( f_n/m \) or \( g_n/m \) for \( F_0/m \). By -writing each factor \( 2n\pi t/\tau \) as \( n\omega t \), with \( \omega\equiv -2\pi/\tau \), +The code here shows a typical example of such a square wave generated using the functionality included in the scipy Python package. We have used a period of \( \tau=0.2 \). + +
+ + +
import numpy as np
+import math
+from scipy import signal
+import matplotlib.pyplot as plt
+
+# number of points
+n = 500
+# start and final times
+t0 = 0.0
+tn = 1.0
+# Period
+t = np.linspace(t0, tn, n, endpoint=False)
+SqrSignal = np.zeros(n)
+SqrSignal = 1.0+signal.square(2*np.pi*5*t)
+plt.plot(t, SqrSignal)
+plt.ylim(-0.5, 2.5)
+plt.show()
++For the sinusoidal example the +period is \( \tau=2\pi/\omega \). However, higher harmonics can also +satisfy the periodicity requirement. In general, any force that +satisfies the periodicity requirement can be expressed as a sum over +harmonics, $$ \begin{equation} -\tag{23} -F(t)=\frac{f_0}{2}+\sum_{n>0}f_n\cos(n\omega t)+g_n\sin(n\omega t). +F(t)=\frac{f_0}{2}+\sum_{n>0} f_n\cos(2n\pi t/\tau)+g_n\sin(2n\pi t/\tau). +\tag{22} \end{equation} $$ -
-The solutions for \( x(t) \) then come from replacing \( \omega \) with -\( n\omega \) for each term in the particular solution, - -$$ -\begin{eqnarray} -x_p(t)&=&\frac{f_0}{2k}+\sum_{n>0} \alpha_n\cos(n\omega t-\delta_n)+\beta_n\sin(n\omega t-\delta_n),\\ -\nonumber -\alpha_n&=&\frac{f_n/m}{\sqrt{((n\omega)^2-\omega_0^2)+4\beta^2n^2\omega^2}},\\ -\nonumber -\beta_n&=&\frac{g_n/m}{\sqrt{((n\omega)^2-\omega_0^2)+4\beta^2n^2\omega^2}},\\ -\nonumber -\delta_n&=&\tan^{-1}\left(\frac{2\beta n\omega}{\omega_0^2-n^2\omega^2}\right). -\end{eqnarray} -$$ -
@@ -460,7 +470,7 @@ $$
-Because the forces have been applied for a long time, any non-zero -damping eliminates the homogenous parts of the solution, so one need -only consider the particular solution for each \( n \). - -
-The problem is considered solved if one can find expressions for the -coefficients \( f_n \) and \( g_n \), even though the solutions are expressed -as an infinite sum. The coefficients can be extracted from the -function \( F(t) \) by - -$$ -\begin{eqnarray} -\tag{24} -f_n&=&\frac{2}{\tau}\int_{-\tau/2}^{\tau/2} dt~F(t)\cos(2n\pi t/\tau),\\ -\nonumber -g_n&=&\frac{2}{\tau}\int_{-\tau/2}^{\tau/2} dt~F(t)\sin(2n\pi t/\tau). -\end{eqnarray} -$$ - -
-To check the consistency of these expressions and to verify -Eq. (24), one can insert the expansion of \( F(t) \) in -Eq. (23) into the expression for the coefficients in -Eq. (24) and see whether - -$$ -\begin{eqnarray} -f_n&=?&\frac{2}{\tau}\int_{-\tau/2}^{\tau/2} dt~\left\{ -\frac{f_0}{2}+\sum_{m>0}f_m\cos(m\omega t)+g_m\sin(m\omega t) -\right\}\cos(n\omega t). -\end{eqnarray} -$$ - -
-Immediately, one can throw away all the terms with \( g_m \) because they -convolute an even and an odd function. The term with \( f_0/2 \) -disappears because \( \cos(n\omega t) \) is equally positive and negative -over the interval and will integrate to zero. For all the terms -\( f_m\cos(m\omega t) \) appearing in the sum, one can use angle addition -formulas to see that \( \cos(m\omega t)\cos(n\omega -t)=(1/2)(\cos[(m+n)\omega t]+\cos[(m-n)\omega t] \). This will integrate -to zero unless \( m=n \). In that case the \( m=n \) term gives +We can write down the answer for +\( x_{pn}(t) \), by substituting \( f_n/m \) or \( g_n/m \) for \( F_0/m \). By +writing each factor \( 2n\pi t/\tau \) as \( n\omega t \), with \( \omega\equiv +2\pi/\tau \), $$ \begin{equation} -\int_{-\tau/2}^{\tau/2}dt~\cos^2(m\omega t)=\frac{\tau}{2}, -\tag{25} +\tag{23} +F(t)=\frac{f_0}{2}+\sum_{n>0}f_n\cos(n\omega t)+g_n\sin(n\omega t). \end{equation} $$
-and +The solutions for \( x(t) \) then come from replacing \( \omega \) with +\( n\omega \) for each term in the particular solution, $$ \begin{eqnarray} -f_n&=?&\frac{2}{\tau}\int_{-\tau/2}^{\tau/2} dt~f_n/2\\ +x_p(t)&=&\frac{f_0}{2k}+\sum_{n>0} \alpha_n\cos(n\omega t-\delta_n)+\beta_n\sin(n\omega t-\delta_n),\\ \nonumber -&=&f_n~\checkmark. +\alpha_n&=&\frac{f_n/m}{\sqrt{((n\omega)^2-\omega_0^2)+4\beta^2n^2\omega^2}},\\ +\nonumber +\beta_n&=&\frac{g_n/m}{\sqrt{((n\omega)^2-\omega_0^2)+4\beta^2n^2\omega^2}},\\ +\nonumber +\delta_n&=&\tan^{-1}\left(\frac{2\beta n\omega}{\omega_0^2-n^2\omega^2}\right). \end{eqnarray} $$ -
-The same method can be used to check for the consistency of \( g_n \). -
@@ -496,7 +461,7 @@ The same method can be used to check for the consistency of \( g_n \).
-The code here uses the Fourier series applied to a -square wave signal. The code here -visualizes the various approximations given by Fourier series compared -with a square wave with period \( T=0.2 \) (dimensionless time), width \( 0.1 \) and max value of the force \( F=2 \). We -see that when we increase the number of components in the Fourier -series, the Fourier series approximation gets closer and closer to the -square wave signal. +Because the forces have been applied for a long time, any non-zero +damping eliminates the homogenous parts of the solution, so one need +only consider the particular solution for each \( n \).
+The problem is considered solved if one can find expressions for the +coefficients \( f_n \) and \( g_n \), even though the solutions are expressed +as an infinite sum. The coefficients can be extracted from the +function \( F(t) \) by - -
import numpy as np
-import math
-from scipy import signal
-import matplotlib.pyplot as plt
+$$
+\begin{eqnarray}
+\tag{24}
+f_n&=&\frac{2}{\tau}\int_{-\tau/2}^{\tau/2} dt~F(t)\cos(2n\pi t/\tau),\\
+\nonumber
+g_n&=&\frac{2}{\tau}\int_{-\tau/2}^{\tau/2} dt~F(t)\sin(2n\pi t/\tau).
+\end{eqnarray}
+$$
+
+
+To check the consistency of these expressions and to verify
+Eq. (24), one can insert the expansion of \( F(t) \) in
+Eq. (23) into the expression for the coefficients in
+Eq. (24) and see whether
+
+$$
+\begin{eqnarray}
+f_n&=?&\frac{2}{\tau}\int_{-\tau/2}^{\tau/2} dt~\left\{
+\frac{f_0}{2}+\sum_{m>0}f_m\cos(m\omega t)+g_m\sin(m\omega t)
+\right\}\cos(n\omega t).
+\end{eqnarray}
+$$
+
+
+Immediately, one can throw away all the terms with \( g_m \) because they
+convolute an even and an odd function. The term with \( f_0/2 \)
+disappears because \( \cos(n\omega t) \) is equally positive and negative
+over the interval and will integrate to zero. For all the terms
+\( f_m\cos(m\omega t) \) appearing in the sum, one can use angle addition
+formulas to see that \( \cos(m\omega t)\cos(n\omega
+t)=(1/2)(\cos[(m+n)\omega t]+\cos[(m-n)\omega t] \). This will integrate
+to zero unless \( m=n \). In that case the \( m=n \) term gives
+
+$$
+\begin{equation}
+\int_{-\tau/2}^{\tau/2}dt~\cos^2(m\omega t)=\frac{\tau}{2},
+\tag{25}
+\end{equation}
+$$
+
+
+and
+
+$$
+\begin{eqnarray}
+f_n&=?&\frac{2}{\tau}\int_{-\tau/2}^{\tau/2} dt~f_n/2\\
+\nonumber
+&=&f_n~\checkmark.
+\end{eqnarray}
+$$
+
+
+The same method can be used to check for the consistency of \( g_n \).
-# number of points
-n = 500
-# start and final times
-t0 = 0.0
-tn = 1.0
-# Period
-T =0.2
-# Max value of square signal
-Fmax= 2.0
-# Width of signal
-Width = 0.1
-t = np.linspace(t0, tn, n, endpoint=False)
-SqrSignal = np.zeros(n)
-FourierSeriesSignal = np.zeros(n)
-SqrSignal = 1.0+signal.square(2*np.pi*5*t+np.pi*Width/T)
-a0 = Fmax*Width/T
-FourierSeriesSignal = a0
-Factor = 2.0*Fmax/np.pi
-for i in range(1,500):
- FourierSeriesSignal += Factor/(i)*np.sin(np.pi*i*Width/T)*np.cos(i*t*2*np.pi/T)
-plt.plot(t, SqrSignal)
-plt.plot(t, FourierSeriesSignal)
-plt.ylim(-0.5, 2.5)
-plt.show()
-
@@ -473,7 +497,7 @@ plt.show()
-More text will be added here +The code here uses the Fourier series applied to a +square wave signal. The code here +visualizes the various approximations given by Fourier series compared +with a square wave with period \( T=0.2 \) (dimensionless time), width \( 0.1 \) and max value of the force \( F=2 \). We +see that when we increase the number of components in the Fourier +series, the Fourier series approximation gets closer and closer to the +square wave signal. +
+ + +
import numpy as np
+import math
+from scipy import signal
+import matplotlib.pyplot as plt
+
+# number of points
+n = 500
+# start and final times
+t0 = 0.0
+tn = 1.0
+# Period
+T =0.2
+# Max value of square signal
+Fmax= 2.0
+# Width of signal
+Width = 0.1
+t = np.linspace(t0, tn, n, endpoint=False)
+SqrSignal = np.zeros(n)
+FourierSeriesSignal = np.zeros(n)
+SqrSignal = 1.0+signal.square(2*np.pi*5*t+np.pi*Width/T)
+a0 = Fmax*Width/T
+FourierSeriesSignal = a0
+Factor = 2.0*Fmax/np.pi
+for i in range(1,500):
+ FourierSeriesSignal += Factor/(i)*np.sin(np.pi*i*Width/T)*np.cos(i*t*2*np.pi/T)
+plt.plot(t, SqrSignal)
+plt.plot(t, FourierSeriesSignal)
+plt.ylim(-0.5, 2.5)
+plt.show()
+
@@ -434,7 +474,7 @@ More text will be added here
-In feilds like signal processing (and imaging as well), one designs -so-called filters. These filters are defined by the convolutions and -are often hand-crafted. One may specify filters for smoothing, edge -detection, frequency reshaping, and similar operations. However with -neural networks the idea is to automatically learn the filters and use -many of them in conjunction with non-linear operations (activation -functions). - -
-As an example consider a neural network operating on sound sequence -data. Assume that we an input vector \( \boldsymbol{x} \) of length \( d=10^6 \). We -construct then a neural network with onle hidden layer only with -\( 10^4 \) nodes. This means that we will have a weight matrix with -\( 10^4\times 10^6=10^{10} \) weights to be determined, together with \( 10^4 \) biases. - -
-Assume furthermore that we have an output layer which is meant to train whether the sound sequence represents a human voice (true) or something else (false). -It means that we have only one output node. But since this output node connects to \( 10^4 \) nodes in the hidden layer, there are in total \( 10^4 \) weights to be determined for the output layer, plus one bias. In total we have +We often use convolutions over more than one dimension at a time. If +we have a two-dimensional image \( I \) as input, we can have a filter +defined by a two-dimensional kernel \( K \). This leads to an output \( S \) $$ -\mathrm{NumberParameters}=10^{10}+10^4+10^4+1 \approx 10^{10}, +S_(i,j)=(I * K)(i,j) = \sum_m\sum_n I(m,n)K(i-m,j-n). $$ -that is ten billion parameters to determine. +
+Convolution is a commutatitave process, which means we can rewrite this equation as +$$ +S_(i,j)=(I * K)(i,j) = \sum_m\sum_n I(i-m,j-n)K(m,n). +$$ + +
+Normally the latter is more straightforward to implement in a machine elarning library since there is less variation in the range of values of \( m \) and \( n \).
@@ -457,7 +450,7 @@ that is ten billion parameters to determine.
-In today’s architecture one can train such neural networks, however -this is a huge number of parameters for the task at hand. In general, -it is a very wasteful and inefficient use of dense matrices as -parameters. Just as importantly, such trained network parameters are -very specific for the type of input data on which they were trained -and the network is not likely to generalize easily to variations in -the input. - -
-The main principles that justify convolutions is locality of -information and repetion of patterns within the signal. Sound samples -of the input in adjacent spots are much more likely to affect each -other than those that are very far away. Similarly, sounds are -repeated in multiple times in the signal. While slightly simplistic, -reasoning about such a sound example demonstrates this. The same -principles then apply to images and other similar data. +Many deep learning libraries implement cross-correlation instead of convolution +$$ +S_(i,j)=(I * K)(i,j) = \sum_m\sum_n I(i+m,j-+)K(m,n). +$$
@@ -449,7 +438,7 @@ principles then apply to images and other similar data.
-As discussed above, CNNs are neural networks built from the assumption that the inputs -to the network are 2D images. This is important because the number of features or pixels in images -grows very fast with the image size, and an enormous number of weights and biases are needed in order to build an accurate network. +In feilds like signal processing (and imaging as well), one designs +so-called filters. These filters are defined by the convolutions and +are often hand-crafted. One may specify filters for smoothing, edge +detection, frequency reshaping, and similar operations. However with +neural networks the idea is to automatically learn the filters and use +many of them in conjunction with non-linear operations (activation +functions).
-As before, we still have our input, a hidden layer and an output. What's novel about convolutional networks -are the convolutional and pooling layers stacked in pairs between the input and the hidden layer. -In addition, the data is no longer represented as a 2D feature matrix, instead each input is a number of 2D -matrices, typically 1 for each color dimension (Red, Green, Blue). +As an example consider a neural network operating on sound sequence +data. Assume that we an input vector \( \boldsymbol{x} \) of length \( d=10^6 \). We +construct then a neural network with onle hidden layer only with +\( 10^4 \) nodes. This means that we will have a weight matrix with +\( 10^4\times 10^6=10^{10} \) weights to be determined, together with \( 10^4 \) biases. + +
+Assume furthermore that we have an output layer which is meant to train whether the sound sequence represents a human voice (true) or something else (false). +It means that we have only one output node. But since this output node connects to \( 10^4 \) nodes in the hidden layer, there are in total \( 10^4 \) weights to be determined for the output layer, plus one bias. In total we have + +$$ +\mathrm{NumberParameters}=10^{10}+10^4+10^4+1 \approx 10^{10}, +$$ + +that is ten billion parameters to determine.
@@ -442,7 +458,7 @@ matrices, typically 1 for each color dimension (Red, Green, Blue).
-It means that to represent the entire -dataset of images, we require a 4D matrix or tensor. This tensor has the dimensions: -$$ -(n_{inputs},\, n_{pixels, width},\, n_{pixels, height},\, depth) . -$$ +In today’s architecture one can train such neural networks, however +this is a huge number of parameters for the task at hand. In general, +it is a very wasteful and inefficient use of dense matrices as +parameters. Just as importantly, such trained network parameters are +very specific for the type of input data on which they were trained +and the network is not likely to generalize easily to variations in +the input. + +
+The main principles that justify convolutions is locality of +information and repetion of patterns within the signal. Sound samples +of the input in adjacent spots are much more likely to affect each +other than those that are very far away. Similarly, sounds are +repeated in multiple times in the signal. While slightly simplistic, +reasoning about such a sound example demonstrates this. The same +principles then apply to images and other similar data.
@@ -438,7 +450,7 @@ $$
-The MNIST dataset consists of grayscale images with a pixel size of -\( 28\times 28 \), meaning we require \( 28 \times 28 = 724 \) weights to each -neuron in the first hidden layer. +As discussed above, CNNs are neural networks built from the assumption that the inputs +to the network are 2D images. This is important because the number of features or pixels in images +grows very fast with the image size, and an enormous number of weights and biases are needed in order to build an accurate network.
-If we were to analyze images of size \( 128\times 128 \) we would require -\( 128 \times 128 = 16384 \) weights to each neuron. Even worse if we were -dealing with color images, as most images are, we have an image matrix -of size \( 128\times 128 \) for each color dimension (Red, Green, Blue), -meaning 3 times the number of weights \( = 49152 \) are required for every -single neuron in the first hidden layer. +As before, we still have our input, a hidden layer and an output. What's novel about convolutional networks +are the convolutional and pooling layers stacked in pairs between the input and the hidden layer. +In addition, the data is no longer represented as a 2D feature matrix, instead each input is a number of 2D +matrices, typically 1 for each color dimension (Red, Green, Blue).
@@ -444,7 +443,7 @@ single neuron in the first hidden layer.
-Images typically have strong local correlations, meaning that a small -part of the image varies little from its neighboring regions. If for -example we have an image of a blue car, we can roughly assume that a -small blue part of the image is surrounded by other blue regions. - -
-Therefore, instead of connecting every single pixel to a neuron in the -first hidden layer, as we have previously done with deep neural -networks, we can instead connect each neuron to a small part of the -image (in all 3 RGB depth dimensions). The size of each small area is -fixed, and known as a receptive. +It means that to represent the entire +dataset of images, we require a 4D matrix or tensor. This tensor has the dimensions: +$$ +(n_{inputs},\, n_{pixels, width},\, n_{pixels, height},\, depth) . +$$
@@ -444,7 +439,7 @@ fixed, and known as a 84
- + -
-A convolution is performed on the image which outputs -a 3D volume of neurons. The weights to the input are arranged in a number of 2D matrices, known as filters. +The MNIST dataset consists of grayscale images with a pixel size of +\( 28\times 28 \), meaning we require \( 28 \times 28 = 724 \) weights to each +neuron in the first hidden layer.
-Each filter slides along the input image, taking the dot product -between each small part of the image and the filter, in all depth -dimensions. This is then passed through a non-linear function, -typically the Rectified Linear (ReLu) function, which serves as the -activation of the neurons in the first convolutional layer. This is -further passed through a pooling layer, which reduces the size of the -convolutional layer, e.g. by taking the maximum or average across some -small regions, and this serves as input to the next convolutional -layer. +If we were to analyze images of size \( 128\times 128 \) we would require +\( 128 \times 128 = 16384 \) weights to each neuron. Even worse if we were +dealing with color images, as most images are, we have an image matrix +of size \( 128\times 128 \) for each color dimension (Red, Green, Blue), +meaning 3 times the number of weights \( = 49152 \) are required for every +single neuron in the first hidden layer.
@@ -448,7 +445,7 @@ layer.
-By systematically reducing the size of the input volume, through -convolution and pooling, the network should create representations of -small parts of the input, and then from them assemble representations -of larger areas. The final pooling layer is flattened to serve as -input to a hidden layer, such that each neuron in the final pooling -layer is connected to every single neuron in the hidden layer. This -then serves as input to the output layer, e.g. a softmax output for -classification. +Images typically have strong local correlations, meaning that a small +part of the image varies little from its neighboring regions. If for +example we have an image of a blue car, we can roughly assume that a +small blue part of the image is surrounded by other blue regions. + +
+Therefore, instead of connecting every single pixel to a neuron in the +first hidden layer, as we have previously done with deep neural +networks, we can instead connect each neuron to a small part of the +image (in all 3 RGB depth dimensions). The size of each small area is +fixed, and known as a receptive.
@@ -441,7 +445,7 @@ classification.
- + + +
+A convolution is performed on the image which outputs +a 3D volume of neurons. The weights to the input are arranged in a number of 2D matrices, known as filters. - -
# import necessary packages
-import numpy as np
-import matplotlib.pyplot as plt
-from sklearn import datasets
+
+Each filter slides along the input image, taking the dot product
+between each small part of the image and the filter, in all depth
+dimensions. This is then passed through a non-linear function,
+typically the Rectified Linear (ReLu) function, which serves as the
+activation of the neurons in the first convolutional layer. This is
+further passed through a pooling layer, which reduces the size of the
+convolutional layer, e.g. by taking the maximum or average across some
+small regions, and this serves as input to the next convolutional
+layer.
-
-# ensure the same random numbers appear every time
-np.random.seed(0)
-
-# display images in notebook
-%matplotlib inline
-plt.rcParams['figure.figsize'] = (12,12)
-
-
-# download MNIST dataset
-digits = datasets.load_digits()
-
-# define inputs and labels
-inputs = digits.images
-labels = digits.target
-
-# RGB images have a depth of 3
-# our images are grayscale so they should have a depth of 1
-inputs = inputs[:,:,:,np.newaxis]
-
-print("inputs = (n_inputs, pixel_width, pixel_height, depth) = " + str(inputs.shape))
-print("labels = (n_inputs) = " + str(labels.shape))
-
-
-# choose some random images to display
-n_inputs = len(inputs)
-indices = np.arange(n_inputs)
-random_indices = np.random.choice(indices, size=5)
-
-for i, image in enumerate(digits.images[random_indices]):
- plt.subplot(1, 5, i+1)
- plt.axis('off')
- plt.imshow(image, cmap=plt.cm.gray_r, interpolation='nearest')
- plt.title("Label: %d" % digits.target[random_indices[i]])
-plt.show()
-
@@ -474,7 +449,7 @@ plt.show()
+By systematically reducing the size of the input volume, through +convolution and pooling, the network should create representations of +small parts of the input, and then from them assemble representations +of larger areas. The final pooling layer is flattened to serve as +input to a hidden layer, such that each neuron in the final pooling +layer is connected to every single neuron in the hidden layer. This +then serves as input to the output layer, e.g. a softmax output for +classification. - -
from tensorflow.keras import datasets, layers, models
-from tensorflow.keras.layers import Input
-from tensorflow.keras.models import Sequential #This allows appending layers to existing models
-from tensorflow.keras.layers import Dense #This allows defining the characteristics of a particular layer
-from tensorflow.keras import optimizers #This allows using whichever optimiser we want (sgd,adam,RMSprop)
-from tensorflow.keras import regularizers #This allows using whichever regularizer we want (l1,l2,l1_l2)
-from tensorflow.keras.utils import to_categorical #This allows using categorical cross entropy as the cost function
-#from tensorflow.keras import Conv2D
-#from tensorflow.keras import MaxPooling2D
-#from tensorflow.keras import Flatten
-
-from sklearn.model_selection import train_test_split
-
-# representation of labels
-labels = to_categorical(labels)
-
-# split into train and test data
-# one-liner from scikit-learn library
-train_size = 0.8
-test_size = 1 - train_size
-X_train, X_test, Y_train, Y_test = train_test_split(inputs, labels, train_size=train_size,
- test_size=test_size)
-
@@ -455,6 +441,8 @@ X_train, X_test, Y_train, Y_test = train_tes
- - -
-
def create_convolutional_neural_network_keras(input_shape, receptive_field,
- n_filters, n_neurons_connected, n_categories,
- eta, lmbd):
- model = Sequential()
- model.add(layers.Conv2D(n_filters, (receptive_field, receptive_field), input_shape=input_shape, padding='same',
- activation='relu', kernel_regularizer=regularizers.l2(lmbd)))
- model.add(layers.MaxPooling2D(pool_size=(2, 2)))
- model.add(layers.Flatten())
- model.add(layers.Dense(n_neurons_connected, activation='relu', kernel_regularizer=regularizers.l2(lmbd)))
- model.add(layers.Dense(n_categories, activation='softmax', kernel_regularizer=regularizers.l2(lmbd)))
-
- sgd = optimizers.SGD(lr=eta)
- model.compile(loss='categorical_crossentropy', optimizer=sgd, metrics=['accuracy'])
-
- return model
+# import necessary packages
+import numpy as np
+import matplotlib.pyplot as plt
+from sklearn import datasets
-epochs = 100
-batch_size = 100
-input_shape = X_train.shape[1:4]
-receptive_field = 3
-n_filters = 10
-n_neurons_connected = 50
-n_categories = 10
-eta_vals = np.logspace(-5, 1, 7)
-lmbd_vals = np.logspace(-5, 1, 7)
+# ensure the same random numbers appear every time
+np.random.seed(0)
+
+# display images in notebook
+%matplotlib inline
+plt.rcParams['figure.figsize'] = (12,12)
+
+
+# download MNIST dataset
+digits = datasets.load_digits()
+
+# define inputs and labels
+inputs = digits.images
+labels = digits.target
+
+# RGB images have a depth of 3
+# our images are grayscale so they should have a depth of 1
+inputs = inputs[:,:,:,np.newaxis]
+
+print("inputs = (n_inputs, pixel_width, pixel_height, depth) = " + str(inputs.shape))
+print("labels = (n_inputs) = " + str(labels.shape))
+
+
+# choose some random images to display
+n_inputs = len(inputs)
+indices = np.arange(n_inputs)
+random_indices = np.random.choice(indices, size=5)
+
+for i, image in enumerate(digits.images[random_indices]):
+ plt.subplot(1, 5, i+1)
+ plt.axis('off')
+ plt.imshow(image, cmap=plt.cm.gray_r, interpolation='nearest')
+ plt.title("Label: %d" % digits.target[random_indices[i]])
+plt.show()
@@ -459,6 +473,9 @@ lmbd_vals = np.
87
88
89
+ 90
+ ...
+ 91
»
diff --git a/doc/pub/week42/html/._week42-bs081.html b/doc/pub/week42/html/._week42-bs081.html
index a2818f109..75dc2d339 100644
--- a/doc/pub/week42/html/._week42-bs081.html
+++ b/doc/pub/week42/html/._week42-bs081.html
@@ -172,6 +172,7 @@ Automatically generated HTML file from DocOnce source
2,
None,
'convolutional-neural-networks-recognizing-images'),
+ ('What is the Difference', 2, None, 'what-is-the-difference'),
('Neural Networks vs CNNs', 2, None, 'neural-networks-vs-cnns'),
('Why CNNS for images, sound files, medical images from CT scans '
'etc?',
@@ -221,10 +222,8 @@ Automatically generated HTML file from DocOnce source
2,
None,
'final-words-on-fourier-transforms'),
- ('Convolution Examples: Probability Theory',
- 2,
- None,
- 'convolution-examples-probability-theory'),
+ ('Two-dimensional Objects', 2, None, 'two-dimensional-objects'),
+ ('Cross-Correlation', 2, None, 'cross-correlation'),
('More on Dimensionalities', 2, None, 'more-on-dimensionalities'),
('Further Dimensionality Remarks',
2,
@@ -351,43 +350,45 @@ MathJax.Hub.Config({
Solving the wave equation - the full program using Autograd
Resources on differential equations and deep learning
Convolutional Neural Networks (recognizing images)
- Neural Networks vs CNNs
- Why CNNS for images, sound files, medical images from CT scans etc?
- Regular NNs don’t scale well to full images
- 3D volumes of neurons
- Layers used to build CNNs
- Transforming images
- CNNs in brief
- Mathematics of CNNs
- Convolution Examples: Polynomial multiplication
- Efficient Polynomial Multiplication
- A more efficient way of coding the above Convolution
- Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
- Principle of Superposition
- Simple Code Example
- Wrapping up Fourier transforms
- Finding the Coefficients
- Final words on Fourier Transforms
- Convolution Examples: Probability Theory
- More on Dimensionalities
- Further Dimensionality Remarks
- CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
- Setting it up
- The MNIST dataset again
- Strong correlations
- Layers of a CNN
- Systematic reduction
- Prerequisites: Collect and pre-process data
- Importing Keras and Tensorflow
- Running with Keras
- Final part
- Final visualization
- The CIFAR01 data set
- Verifying the data set
- Set up the model
- Add Dense layers on top
- Compile and train the model
- Finally, evaluate the model
+ What is the Difference
+ Neural Networks vs CNNs
+ Why CNNS for images, sound files, medical images from CT scans etc?
+ Regular NNs don’t scale well to full images
+ 3D volumes of neurons
+ Layers used to build CNNs
+ Transforming images
+ CNNs in brief
+ Mathematics of CNNs
+ Convolution Examples: Polynomial multiplication
+ Efficient Polynomial Multiplication
+ A more efficient way of coding the above Convolution
+ Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
+ Principle of Superposition
+ Simple Code Example
+ Wrapping up Fourier transforms
+ Finding the Coefficients
+ Final words on Fourier Transforms
+ Two-dimensional Objects
+ Cross-Correlation
+ More on Dimensionalities
+ Further Dimensionality Remarks
+ CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
+ Setting it up
+ The MNIST dataset again
+ Strong correlations
+ Layers of a CNN
+ Systematic reduction
+ Prerequisites: Collect and pre-process data
+ Importing Keras and Tensorflow
+ Running with Keras
+ Final part
+ Final visualization
+ The CIFAR01 data set
+ Verifying the data set
+ Set up the model
+ Add Dense layers on top
+ Compile and train the model
+ Finally, evaluate the model
@@ -403,27 +404,32 @@ MathJax.Hub.Config({
-Final part
-
+Importing Keras and Tensorflow
-
CNN_keras = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)
-
-for i, eta in enumerate(eta_vals):
- for j, lmbd in enumerate(lmbd_vals):
- CNN = create_convolutional_neural_network_keras(input_shape, receptive_field,
- n_filters, n_neurons_connected, n_categories,
- eta, lmbd)
- CNN.fit(X_train, Y_train, epochs=epochs, batch_size=batch_size, verbose=0)
- scores = CNN.evaluate(X_test, Y_test)
-
- CNN_keras[i][j] = CNN
-
- print("Learning rate = ", eta)
- print("Lambda = ", lmbd)
- print("Test accuracy: %.3f" % scores[1])
- print()
+from tensorflow.keras import datasets, layers, models
+from tensorflow.keras.layers import Input
+from tensorflow.keras.models import Sequential #This allows appending layers to existing models
+from tensorflow.keras.layers import Dense #This allows defining the characteristics of a particular layer
+from tensorflow.keras import optimizers #This allows using whichever optimiser we want (sgd,adam,RMSprop)
+from tensorflow.keras import regularizers #This allows using whichever regularizer we want (l1,l2,l1_l2)
+from tensorflow.keras.utils import to_categorical #This allows using categorical cross entropy as the cost function
+#from tensorflow.keras import Conv2D
+#from tensorflow.keras import MaxPooling2D
+#from tensorflow.keras import Flatten
+
+from sklearn.model_selection import train_test_split
+
+# representation of labels
+labels = to_categorical(labels)
+
+# split into train and test data
+# one-liner from scikit-learn library
+train_size = 0.8
+test_size = 1 - train_size
+X_train, X_test, Y_train, Y_test = train_test_split(inputs, labels, train_size=train_size,
+ test_size=test_size)
@@ -448,6 +454,8 @@ MathJax.Hub.Config({
87
88
89
+ 90
+ 91
»
diff --git a/doc/pub/week42/html/._week42-bs082.html b/doc/pub/week42/html/._week42-bs082.html
index 69633de8d..b0fef9b9a 100644
--- a/doc/pub/week42/html/._week42-bs082.html
+++ b/doc/pub/week42/html/._week42-bs082.html
@@ -172,6 +172,7 @@ Automatically generated HTML file from DocOnce source
2,
None,
'convolutional-neural-networks-recognizing-images'),
+ ('What is the Difference', 2, None, 'what-is-the-difference'),
('Neural Networks vs CNNs', 2, None, 'neural-networks-vs-cnns'),
('Why CNNS for images, sound files, medical images from CT scans '
'etc?',
@@ -221,10 +222,8 @@ Automatically generated HTML file from DocOnce source
2,
None,
'final-words-on-fourier-transforms'),
- ('Convolution Examples: Probability Theory',
- 2,
- None,
- 'convolution-examples-probability-theory'),
+ ('Two-dimensional Objects', 2, None, 'two-dimensional-objects'),
+ ('Cross-Correlation', 2, None, 'cross-correlation'),
('More on Dimensionalities', 2, None, 'more-on-dimensionalities'),
('Further Dimensionality Remarks',
2,
@@ -351,43 +350,45 @@ MathJax.Hub.Config({
Solving the wave equation - the full program using Autograd
Resources on differential equations and deep learning
Convolutional Neural Networks (recognizing images)
- Neural Networks vs CNNs
- Why CNNS for images, sound files, medical images from CT scans etc?
- Regular NNs don’t scale well to full images
- 3D volumes of neurons
- Layers used to build CNNs
- Transforming images
- CNNs in brief
- Mathematics of CNNs
- Convolution Examples: Polynomial multiplication
- Efficient Polynomial Multiplication
- A more efficient way of coding the above Convolution
- Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
- Principle of Superposition
- Simple Code Example
- Wrapping up Fourier transforms
- Finding the Coefficients
- Final words on Fourier Transforms
- Convolution Examples: Probability Theory
- More on Dimensionalities
- Further Dimensionality Remarks
- CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
- Setting it up
- The MNIST dataset again
- Strong correlations
- Layers of a CNN
- Systematic reduction
- Prerequisites: Collect and pre-process data
- Importing Keras and Tensorflow
- Running with Keras
- Final part
- Final visualization
- The CIFAR01 data set
- Verifying the data set
- Set up the model
- Add Dense layers on top
- Compile and train the model
- Finally, evaluate the model
+ What is the Difference
+ Neural Networks vs CNNs
+ Why CNNS for images, sound files, medical images from CT scans etc?
+ Regular NNs don’t scale well to full images
+ 3D volumes of neurons
+ Layers used to build CNNs
+ Transforming images
+ CNNs in brief
+ Mathematics of CNNs
+ Convolution Examples: Polynomial multiplication
+ Efficient Polynomial Multiplication
+ A more efficient way of coding the above Convolution
+ Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
+ Principle of Superposition
+ Simple Code Example
+ Wrapping up Fourier transforms
+ Finding the Coefficients
+ Final words on Fourier Transforms
+ Two-dimensional Objects
+ Cross-Correlation
+ More on Dimensionalities
+ Further Dimensionality Remarks
+ CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
+ Setting it up
+ The MNIST dataset again
+ Strong correlations
+ Layers of a CNN
+ Systematic reduction
+ Prerequisites: Collect and pre-process data
+ Importing Keras and Tensorflow
+ Running with Keras
+ Final part
+ Final visualization
+ The CIFAR01 data set
+ Verifying the data set
+ Set up the model
+ Add Dense layers on top
+ Compile and train the model
+ Finally, evaluate the model
@@ -401,43 +402,39 @@ MathJax.Hub.Config({
-
+
-Final visualization
+Running with Keras
-
# visual representation of grid search
-# uses seaborn heatmap, could probably do this in matplotlib
-import seaborn as sns
+def create_convolutional_neural_network_keras(input_shape, receptive_field,
+ n_filters, n_neurons_connected, n_categories,
+ eta, lmbd):
+ model = Sequential()
+ model.add(layers.Conv2D(n_filters, (receptive_field, receptive_field), input_shape=input_shape, padding='same',
+ activation='relu', kernel_regularizer=regularizers.l2(lmbd)))
+ model.add(layers.MaxPooling2D(pool_size=(2, 2)))
+ model.add(layers.Flatten())
+ model.add(layers.Dense(n_neurons_connected, activation='relu', kernel_regularizer=regularizers.l2(lmbd)))
+ model.add(layers.Dense(n_categories, activation='softmax', kernel_regularizer=regularizers.l2(lmbd)))
+
+ sgd = optimizers.SGD(lr=eta)
+ model.compile(loss='categorical_crossentropy', optimizer=sgd, metrics=['accuracy'])
+
+ return model
-sns.set()
+epochs = 100
+batch_size = 100
+input_shape = X_train.shape[1:4]
+receptive_field = 3
+n_filters = 10
+n_neurons_connected = 50
+n_categories = 10
-train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
-test_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
-
-for i in range(len(eta_vals)):
- for j in range(len(lmbd_vals)):
- CNN = CNN_keras[i][j]
-
- train_accuracy[i][j] = CNN.evaluate(X_train, Y_train)[1]
- test_accuracy[i][j] = CNN.evaluate(X_test, Y_test)[1]
-
-
-fig, ax = plt.subplots(figsize = (10, 10))
-sns.heatmap(train_accuracy, annot=True, ax=ax, cmap="viridis")
-ax.set_title("Training Accuracy")
-ax.set_ylabel("$\eta$")
-ax.set_xlabel("$\lambda$")
-plt.show()
-
-fig, ax = plt.subplots(figsize = (10, 10))
-sns.heatmap(test_accuracy, annot=True, ax=ax, cmap="viridis")
-ax.set_title("Test Accuracy")
-ax.set_ylabel("$\eta$")
-ax.set_xlabel("$\lambda$")
-plt.show()
+eta_vals = np.logspace(-5, 1, 7)
+lmbd_vals = np.logspace(-5, 1, 7)
@@ -461,6 +458,8 @@ plt.show()
87
88
89
+ 90
+ 91
»
diff --git a/doc/pub/week42/html/._week42-bs083.html b/doc/pub/week42/html/._week42-bs083.html
index bd7cebe10..21e2ec214 100644
--- a/doc/pub/week42/html/._week42-bs083.html
+++ b/doc/pub/week42/html/._week42-bs083.html
@@ -172,6 +172,7 @@ Automatically generated HTML file from DocOnce source
2,
None,
'convolutional-neural-networks-recognizing-images'),
+ ('What is the Difference', 2, None, 'what-is-the-difference'),
('Neural Networks vs CNNs', 2, None, 'neural-networks-vs-cnns'),
('Why CNNS for images, sound files, medical images from CT scans '
'etc?',
@@ -221,10 +222,8 @@ Automatically generated HTML file from DocOnce source
2,
None,
'final-words-on-fourier-transforms'),
- ('Convolution Examples: Probability Theory',
- 2,
- None,
- 'convolution-examples-probability-theory'),
+ ('Two-dimensional Objects', 2, None, 'two-dimensional-objects'),
+ ('Cross-Correlation', 2, None, 'cross-correlation'),
('More on Dimensionalities', 2, None, 'more-on-dimensionalities'),
('Further Dimensionality Remarks',
2,
@@ -351,43 +350,45 @@ MathJax.Hub.Config({
Solving the wave equation - the full program using Autograd
Resources on differential equations and deep learning
Convolutional Neural Networks (recognizing images)
- Neural Networks vs CNNs
- Why CNNS for images, sound files, medical images from CT scans etc?
- Regular NNs don’t scale well to full images
- 3D volumes of neurons
- Layers used to build CNNs
- Transforming images
- CNNs in brief
- Mathematics of CNNs
- Convolution Examples: Polynomial multiplication
- Efficient Polynomial Multiplication
- A more efficient way of coding the above Convolution
- Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
- Principle of Superposition
- Simple Code Example
- Wrapping up Fourier transforms
- Finding the Coefficients
- Final words on Fourier Transforms
- Convolution Examples: Probability Theory
- More on Dimensionalities
- Further Dimensionality Remarks
- CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
- Setting it up
- The MNIST dataset again
- Strong correlations
- Layers of a CNN
- Systematic reduction
- Prerequisites: Collect and pre-process data
- Importing Keras and Tensorflow
- Running with Keras
- Final part
- Final visualization
- The CIFAR01 data set
- Verifying the data set
- Set up the model
- Add Dense layers on top
- Compile and train the model
- Finally, evaluate the model
+ What is the Difference
+ Neural Networks vs CNNs
+ Why CNNS for images, sound files, medical images from CT scans etc?
+ Regular NNs don’t scale well to full images
+ 3D volumes of neurons
+ Layers used to build CNNs
+ Transforming images
+ CNNs in brief
+ Mathematics of CNNs
+ Convolution Examples: Polynomial multiplication
+ Efficient Polynomial Multiplication
+ A more efficient way of coding the above Convolution
+ Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
+ Principle of Superposition
+ Simple Code Example
+ Wrapping up Fourier transforms
+ Finding the Coefficients
+ Final words on Fourier Transforms
+ Two-dimensional Objects
+ Cross-Correlation
+ More on Dimensionalities
+ Further Dimensionality Remarks
+ CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
+ Setting it up
+ The MNIST dataset again
+ Strong correlations
+ Layers of a CNN
+ Systematic reduction
+ Prerequisites: Collect and pre-process data
+ Importing Keras and Tensorflow
+ Running with Keras
+ Final part
+ Final visualization
+ The CIFAR01 data set
+ Verifying the data set
+ Set up the model
+ Add Dense layers on top
+ Compile and train the model
+ Finally, evaluate the model
@@ -403,27 +404,27 @@ MathJax.Hub.Config({
-The CIFAR01 data set
-
-
-The CIFAR10 dataset contains 60,000 color images in 10 classes, with
-6,000 images in each class. The dataset is divided into 50,000
-training images and 10,000 testing images. The classes are mutually
-exclusive and there is no overlap between them.
+
Final part
-
import tensorflow as tf
-
-from tensorflow.keras import datasets, layers, models
-import matplotlib.pyplot as plt
-
-# We import the data set
-(train_images, train_labels), (test_images, test_labels) = datasets.cifar10.load_data()
-
-# Normalize pixel values to be between 0 and 1 by dividing by 255.
-train_images, test_images = train_images / 255.0, test_images / 255.0
+CNN_keras = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)
+
+for i, eta in enumerate(eta_vals):
+ for j, lmbd in enumerate(lmbd_vals):
+ CNN = create_convolutional_neural_network_keras(input_shape, receptive_field,
+ n_filters, n_neurons_connected, n_categories,
+ eta, lmbd)
+ CNN.fit(X_train, Y_train, epochs=epochs, batch_size=batch_size, verbose=0)
+ scores = CNN.evaluate(X_test, Y_test)
+
+ CNN_keras[i][j] = CNN
+
+ print("Learning rate = ", eta)
+ print("Lambda = ", lmbd)
+ print("Test accuracy: %.3f" % scores[1])
+ print()
@@ -446,6 +447,8 @@ train_images, test_images = train_images 87
88
89
+ 90
+ 91
»
diff --git a/doc/pub/week42/html/._week42-bs084.html b/doc/pub/week42/html/._week42-bs084.html
index fc782da10..6ebb54a7b 100644
--- a/doc/pub/week42/html/._week42-bs084.html
+++ b/doc/pub/week42/html/._week42-bs084.html
@@ -172,6 +172,7 @@ Automatically generated HTML file from DocOnce source
2,
None,
'convolutional-neural-networks-recognizing-images'),
+ ('What is the Difference', 2, None, 'what-is-the-difference'),
('Neural Networks vs CNNs', 2, None, 'neural-networks-vs-cnns'),
('Why CNNS for images, sound files, medical images from CT scans '
'etc?',
@@ -221,10 +222,8 @@ Automatically generated HTML file from DocOnce source
2,
None,
'final-words-on-fourier-transforms'),
- ('Convolution Examples: Probability Theory',
- 2,
- None,
- 'convolution-examples-probability-theory'),
+ ('Two-dimensional Objects', 2, None, 'two-dimensional-objects'),
+ ('Cross-Correlation', 2, None, 'cross-correlation'),
('More on Dimensionalities', 2, None, 'more-on-dimensionalities'),
('Further Dimensionality Remarks',
2,
@@ -351,43 +350,45 @@ MathJax.Hub.Config({
Solving the wave equation - the full program using Autograd
Resources on differential equations and deep learning
Convolutional Neural Networks (recognizing images)
- Neural Networks vs CNNs
- Why CNNS for images, sound files, medical images from CT scans etc?
- Regular NNs don’t scale well to full images
- 3D volumes of neurons
- Layers used to build CNNs
- Transforming images
- CNNs in brief
- Mathematics of CNNs
- Convolution Examples: Polynomial multiplication
- Efficient Polynomial Multiplication
- A more efficient way of coding the above Convolution
- Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
- Principle of Superposition
- Simple Code Example
- Wrapping up Fourier transforms
- Finding the Coefficients
- Final words on Fourier Transforms
- Convolution Examples: Probability Theory
- More on Dimensionalities
- Further Dimensionality Remarks
- CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
- Setting it up
- The MNIST dataset again
- Strong correlations
- Layers of a CNN
- Systematic reduction
- Prerequisites: Collect and pre-process data
- Importing Keras and Tensorflow
- Running with Keras
- Final part
- Final visualization
- The CIFAR01 data set
- Verifying the data set
- Set up the model
- Add Dense layers on top
- Compile and train the model
- Finally, evaluate the model
+ What is the Difference
+ Neural Networks vs CNNs
+ Why CNNS for images, sound files, medical images from CT scans etc?
+ Regular NNs don’t scale well to full images
+ 3D volumes of neurons
+ Layers used to build CNNs
+ Transforming images
+ CNNs in brief
+ Mathematics of CNNs
+ Convolution Examples: Polynomial multiplication
+ Efficient Polynomial Multiplication
+ A more efficient way of coding the above Convolution
+ Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
+ Principle of Superposition
+ Simple Code Example
+ Wrapping up Fourier transforms
+ Finding the Coefficients
+ Final words on Fourier Transforms
+ Two-dimensional Objects
+ Cross-Correlation
+ More on Dimensionalities
+ Further Dimensionality Remarks
+ CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
+ Setting it up
+ The MNIST dataset again
+ Strong correlations
+ Layers of a CNN
+ Systematic reduction
+ Prerequisites: Collect and pre-process data
+ Importing Keras and Tensorflow
+ Running with Keras
+ Final part
+ Final visualization
+ The CIFAR01 data set
+ Verifying the data set
+ Set up the model
+ Add Dense layers on top
+ Compile and train the model
+ Finally, evaluate the model
@@ -403,27 +404,40 @@ MathJax.Hub.Config({
-Verifying the data set
-
-
-To verify that the dataset looks correct, let's plot the first 25 images from the training set and display the class name below each image.
+
Final visualization
-
class_names = ['airplane', 'automobile', 'bird', 'cat', 'deer',
- 'dog', 'frog', 'horse', 'ship', 'truck']
-
-plt.figure(figsize=(10,10))
-for i in range(25):
- plt.subplot(5,5,i+1)
- plt.xticks([])
- plt.yticks([])
- plt.grid(False)
- plt.imshow(train_images[i], cmap=plt.cm.binary)
- # The CIFAR labels happen to be arrays,
- # which is why you need the extra index
- plt.xlabel(class_names[train_labels[i][0]])
+# visual representation of grid search
+# uses seaborn heatmap, could probably do this in matplotlib
+import seaborn as sns
+
+sns.set()
+
+train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
+test_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
+
+for i in range(len(eta_vals)):
+ for j in range(len(lmbd_vals)):
+ CNN = CNN_keras[i][j]
+
+ train_accuracy[i][j] = CNN.evaluate(X_train, Y_train)[1]
+ test_accuracy[i][j] = CNN.evaluate(X_test, Y_test)[1]
+
+
+fig, ax = plt.subplots(figsize = (10, 10))
+sns.heatmap(train_accuracy, annot=True, ax=ax, cmap="viridis")
+ax.set_title("Training Accuracy")
+ax.set_ylabel("$\eta$")
+ax.set_xlabel("$\lambda$")
+plt.show()
+
+fig, ax = plt.subplots(figsize = (10, 10))
+sns.heatmap(test_accuracy, annot=True, ax=ax, cmap="viridis")
+ax.set_title("Test Accuracy")
+ax.set_ylabel("$\eta$")
+ax.set_xlabel("$\lambda$")
plt.show()
@@ -446,6 +460,8 @@ plt.show()
87
88
89
+ 90
+ 91
»
diff --git a/doc/pub/week42/html/._week42-bs085.html b/doc/pub/week42/html/._week42-bs085.html
index f0bc45ba2..de3a7aae0 100644
--- a/doc/pub/week42/html/._week42-bs085.html
+++ b/doc/pub/week42/html/._week42-bs085.html
@@ -172,6 +172,7 @@ Automatically generated HTML file from DocOnce source
2,
None,
'convolutional-neural-networks-recognizing-images'),
+ ('What is the Difference', 2, None, 'what-is-the-difference'),
('Neural Networks vs CNNs', 2, None, 'neural-networks-vs-cnns'),
('Why CNNS for images, sound files, medical images from CT scans '
'etc?',
@@ -221,10 +222,8 @@ Automatically generated HTML file from DocOnce source
2,
None,
'final-words-on-fourier-transforms'),
- ('Convolution Examples: Probability Theory',
- 2,
- None,
- 'convolution-examples-probability-theory'),
+ ('Two-dimensional Objects', 2, None, 'two-dimensional-objects'),
+ ('Cross-Correlation', 2, None, 'cross-correlation'),
('More on Dimensionalities', 2, None, 'more-on-dimensionalities'),
('Further Dimensionality Remarks',
2,
@@ -351,43 +350,45 @@ MathJax.Hub.Config({
Solving the wave equation - the full program using Autograd
Resources on differential equations and deep learning
Convolutional Neural Networks (recognizing images)
- Neural Networks vs CNNs
- Why CNNS for images, sound files, medical images from CT scans etc?
- Regular NNs don’t scale well to full images
- 3D volumes of neurons
- Layers used to build CNNs
- Transforming images
- CNNs in brief
- Mathematics of CNNs
- Convolution Examples: Polynomial multiplication
- Efficient Polynomial Multiplication
- A more efficient way of coding the above Convolution
- Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
- Principle of Superposition
- Simple Code Example
- Wrapping up Fourier transforms
- Finding the Coefficients
- Final words on Fourier Transforms
- Convolution Examples: Probability Theory
- More on Dimensionalities
- Further Dimensionality Remarks
- CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
- Setting it up
- The MNIST dataset again
- Strong correlations
- Layers of a CNN
- Systematic reduction
- Prerequisites: Collect and pre-process data
- Importing Keras and Tensorflow
- Running with Keras
- Final part
- Final visualization
- The CIFAR01 data set
- Verifying the data set
- Set up the model
- Add Dense layers on top
- Compile and train the model
- Finally, evaluate the model
+ What is the Difference
+ Neural Networks vs CNNs
+ Why CNNS for images, sound files, medical images from CT scans etc?
+ Regular NNs don’t scale well to full images
+ 3D volumes of neurons
+ Layers used to build CNNs
+ Transforming images
+ CNNs in brief
+ Mathematics of CNNs
+ Convolution Examples: Polynomial multiplication
+ Efficient Polynomial Multiplication
+ A more efficient way of coding the above Convolution
+ Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
+ Principle of Superposition
+ Simple Code Example
+ Wrapping up Fourier transforms
+ Finding the Coefficients
+ Final words on Fourier Transforms
+ Two-dimensional Objects
+ Cross-Correlation
+ More on Dimensionalities
+ Further Dimensionality Remarks
+ CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
+ Setting it up
+ The MNIST dataset again
+ Strong correlations
+ Layers of a CNN
+ Systematic reduction
+ Prerequisites: Collect and pre-process data
+ Importing Keras and Tensorflow
+ Running with Keras
+ Final part
+ Final visualization
+ The CIFAR01 data set
+ Verifying the data set
+ Set up the model
+ Add Dense layers on top
+ Compile and train the model
+ Finally, evaluate the model
@@ -403,31 +404,28 @@ MathJax.Hub.Config({
-Set up the model
+The CIFAR01 data set
-The 6 lines of code below define the convolutional base using a common pattern: a stack of Conv2D and MaxPooling2D layers.
-
-
-As input, a CNN takes tensors of shape (image_height, image_width, color_channels), ignoring the batch size. If you are new to these dimensions, color_channels refers to (R,G,B). In this example, you will configure our CNN to process inputs of shape (32, 32, 3), which is the format of CIFAR images. You can do this by passing the argument input_shape to our first layer.
+The CIFAR10 dataset contains 60,000 color images in 10 classes, with
+6,000 images in each class. The dataset is divided into 50,000
+training images and 10,000 testing images. The classes are mutually
+exclusive and there is no overlap between them.
-
model = models.Sequential()
-model.add(layers.Conv2D(32, (3, 3), activation='relu', input_shape=(32, 32, 3)))
-model.add(layers.MaxPooling2D((2, 2)))
-model.add(layers.Conv2D(64, (3, 3), activation='relu'))
-model.add(layers.MaxPooling2D((2, 2)))
-model.add(layers.Conv2D(64, (3, 3), activation='relu'))
+import tensorflow as tf
-# Let's display the architecture of our model so far.
+from tensorflow.keras import datasets, layers, models
+import matplotlib.pyplot as plt
-model.summary()
+# We import the data set
+(train_images, train_labels), (test_images, test_labels) = datasets.cifar10.load_data()
+
+# Normalize pixel values to be between 0 and 1 by dividing by 255.
+train_images, test_images = train_images / 255.0, test_images / 255.0
-
-You can see that the output of every Conv2D and MaxPooling2D layer is a 3D tensor of shape (height, width, channels). The width and height dimensions tend to shrink as you go deeper in the network. The number of output channels for each Conv2D layer is controlled by the first argument (e.g., 32 or 64). Typically, as the width and height shrink, you can afford (computationally) to add more output channels in each Conv2D layer.
-
@@ -447,6 +445,8 @@ You can see that the output of every Conv2D and MaxPooling2D layer is a 3D tenso
87
88
89
+ 90
+ 91
»
diff --git a/doc/pub/week42/html/._week42-bs086.html b/doc/pub/week42/html/._week42-bs086.html
index 5a0ecc1f7..f40123642 100644
--- a/doc/pub/week42/html/._week42-bs086.html
+++ b/doc/pub/week42/html/._week42-bs086.html
@@ -172,6 +172,7 @@ Automatically generated HTML file from DocOnce source
2,
None,
'convolutional-neural-networks-recognizing-images'),
+ ('What is the Difference', 2, None, 'what-is-the-difference'),
('Neural Networks vs CNNs', 2, None, 'neural-networks-vs-cnns'),
('Why CNNS for images, sound files, medical images from CT scans '
'etc?',
@@ -221,10 +222,8 @@ Automatically generated HTML file from DocOnce source
2,
None,
'final-words-on-fourier-transforms'),
- ('Convolution Examples: Probability Theory',
- 2,
- None,
- 'convolution-examples-probability-theory'),
+ ('Two-dimensional Objects', 2, None, 'two-dimensional-objects'),
+ ('Cross-Correlation', 2, None, 'cross-correlation'),
('More on Dimensionalities', 2, None, 'more-on-dimensionalities'),
('Further Dimensionality Remarks',
2,
@@ -351,43 +350,45 @@ MathJax.Hub.Config({
Solving the wave equation - the full program using Autograd
Resources on differential equations and deep learning
Convolutional Neural Networks (recognizing images)
- Neural Networks vs CNNs
- Why CNNS for images, sound files, medical images from CT scans etc?
- Regular NNs don’t scale well to full images
- 3D volumes of neurons
- Layers used to build CNNs
- Transforming images
- CNNs in brief
- Mathematics of CNNs
- Convolution Examples: Polynomial multiplication
- Efficient Polynomial Multiplication
- A more efficient way of coding the above Convolution
- Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
- Principle of Superposition
- Simple Code Example
- Wrapping up Fourier transforms
- Finding the Coefficients
- Final words on Fourier Transforms
- Convolution Examples: Probability Theory
- More on Dimensionalities
- Further Dimensionality Remarks
- CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
- Setting it up
- The MNIST dataset again
- Strong correlations
- Layers of a CNN
- Systematic reduction
- Prerequisites: Collect and pre-process data
- Importing Keras and Tensorflow
- Running with Keras
- Final part
- Final visualization
- The CIFAR01 data set
- Verifying the data set
- Set up the model
- Add Dense layers on top
- Compile and train the model
- Finally, evaluate the model
+ What is the Difference
+ Neural Networks vs CNNs
+ Why CNNS for images, sound files, medical images from CT scans etc?
+ Regular NNs don’t scale well to full images
+ 3D volumes of neurons
+ Layers used to build CNNs
+ Transforming images
+ CNNs in brief
+ Mathematics of CNNs
+ Convolution Examples: Polynomial multiplication
+ Efficient Polynomial Multiplication
+ A more efficient way of coding the above Convolution
+ Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
+ Principle of Superposition
+ Simple Code Example
+ Wrapping up Fourier transforms
+ Finding the Coefficients
+ Final words on Fourier Transforms
+ Two-dimensional Objects
+ Cross-Correlation
+ More on Dimensionalities
+ Further Dimensionality Remarks
+ CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
+ Setting it up
+ The MNIST dataset again
+ Strong correlations
+ Layers of a CNN
+ Systematic reduction
+ Prerequisites: Collect and pre-process data
+ Importing Keras and Tensorflow
+ Running with Keras
+ Final part
+ Final visualization
+ The CIFAR01 data set
+ Verifying the data set
+ Set up the model
+ Add Dense layers on top
+ Compile and train the model
+ Finally, evaluate the model
@@ -403,30 +404,29 @@ MathJax.Hub.Config({
-Add Dense layers on top
+Verifying the data set
-To complete our model, you will feed the last output tensor from the
-convolutional base (of shape (4, 4, 64)) into one or more Dense layers
-to perform classification. Dense layers take vectors as input (which
-are 1D), while the current output is a 3D tensor. First, you will
-flatten (or unroll) the 3D output to 1D, then add one or more Dense
-layers on top. CIFAR has 10 output classes, so you use a final Dense
-layer with 10 outputs and a softmax activation.
+To verify that the dataset looks correct, let's plot the first 25 images from the training set and display the class name below each image.
-
model.add(layers.Flatten())
-model.add(layers.Dense(64, activation='relu'))
-model.add(layers.Dense(10))
-Here's the complete architecture of our model.
-
-model.summary()
+class_names = ['airplane', 'automobile', 'bird', 'cat', 'deer',
+ 'dog', 'frog', 'horse', 'ship', 'truck']
+
+plt.figure(figsize=(10,10))
+for i in range(25):
+ plt.subplot(5,5,i+1)
+ plt.xticks([])
+ plt.yticks([])
+ plt.grid(False)
+ plt.imshow(train_images[i], cmap=plt.cm.binary)
+ # The CIFAR labels happen to be arrays,
+ # which is why you need the extra index
+ plt.xlabel(class_names[train_labels[i][0]])
+plt.show()
-
-As you can see, our (4, 4, 64) outputs were flattened into vectors of shape (1024) before going through two Dense layers.
-
@@ -445,6 +445,8 @@ As you can see, our (4, 4, 64) outputs were flattened into vectors of shape (102
87
88
89
+ 90
+ 91
»
diff --git a/doc/pub/week42/html/._week42-bs087.html b/doc/pub/week42/html/._week42-bs087.html
index 477ba9eab..15c11f9a5 100644
--- a/doc/pub/week42/html/._week42-bs087.html
+++ b/doc/pub/week42/html/._week42-bs087.html
@@ -172,6 +172,7 @@ Automatically generated HTML file from DocOnce source
2,
None,
'convolutional-neural-networks-recognizing-images'),
+ ('What is the Difference', 2, None, 'what-is-the-difference'),
('Neural Networks vs CNNs', 2, None, 'neural-networks-vs-cnns'),
('Why CNNS for images, sound files, medical images from CT scans '
'etc?',
@@ -221,10 +222,8 @@ Automatically generated HTML file from DocOnce source
2,
None,
'final-words-on-fourier-transforms'),
- ('Convolution Examples: Probability Theory',
- 2,
- None,
- 'convolution-examples-probability-theory'),
+ ('Two-dimensional Objects', 2, None, 'two-dimensional-objects'),
+ ('Cross-Correlation', 2, None, 'cross-correlation'),
('More on Dimensionalities', 2, None, 'more-on-dimensionalities'),
('Further Dimensionality Remarks',
2,
@@ -351,43 +350,45 @@ MathJax.Hub.Config({
Solving the wave equation - the full program using Autograd
Resources on differential equations and deep learning
Convolutional Neural Networks (recognizing images)
- Neural Networks vs CNNs
- Why CNNS for images, sound files, medical images from CT scans etc?
- Regular NNs don’t scale well to full images
- 3D volumes of neurons
- Layers used to build CNNs
- Transforming images
- CNNs in brief
- Mathematics of CNNs
- Convolution Examples: Polynomial multiplication
- Efficient Polynomial Multiplication
- A more efficient way of coding the above Convolution
- Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
- Principle of Superposition
- Simple Code Example
- Wrapping up Fourier transforms
- Finding the Coefficients
- Final words on Fourier Transforms
- Convolution Examples: Probability Theory
- More on Dimensionalities
- Further Dimensionality Remarks
- CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
- Setting it up
- The MNIST dataset again
- Strong correlations
- Layers of a CNN
- Systematic reduction
- Prerequisites: Collect and pre-process data
- Importing Keras and Tensorflow
- Running with Keras
- Final part
- Final visualization
- The CIFAR01 data set
- Verifying the data set
- Set up the model
- Add Dense layers on top
- Compile and train the model
- Finally, evaluate the model
+ What is the Difference
+ Neural Networks vs CNNs
+ Why CNNS for images, sound files, medical images from CT scans etc?
+ Regular NNs don’t scale well to full images
+ 3D volumes of neurons
+ Layers used to build CNNs
+ Transforming images
+ CNNs in brief
+ Mathematics of CNNs
+ Convolution Examples: Polynomial multiplication
+ Efficient Polynomial Multiplication
+ A more efficient way of coding the above Convolution
+ Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
+ Principle of Superposition
+ Simple Code Example
+ Wrapping up Fourier transforms
+ Finding the Coefficients
+ Final words on Fourier Transforms
+ Two-dimensional Objects
+ Cross-Correlation
+ More on Dimensionalities
+ Further Dimensionality Remarks
+ CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
+ Setting it up
+ The MNIST dataset again
+ Strong correlations
+ Layers of a CNN
+ Systematic reduction
+ Prerequisites: Collect and pre-process data
+ Importing Keras and Tensorflow
+ Running with Keras
+ Final part
+ Final visualization
+ The CIFAR01 data set
+ Verifying the data set
+ Set up the model
+ Add Dense layers on top
+ Compile and train the model
+ Finally, evaluate the model
@@ -403,18 +404,31 @@ MathJax.Hub.Config({
-Compile and train the model
+Set up the model
+
+
+The 6 lines of code below define the convolutional base using a common pattern: a stack of Conv2D and MaxPooling2D layers.
+
+
+As input, a CNN takes tensors of shape (image_height, image_width, color_channels), ignoring the batch size. If you are new to these dimensions, color_channels refers to (R,G,B). In this example, you will configure our CNN to process inputs of shape (32, 32, 3), which is the format of CIFAR images. You can do this by passing the argument input_shape to our first layer.
-
model.compile(optimizer='adam',
- loss=tf.keras.losses.SparseCategoricalCrossentropy(from_logits=True),
- metrics=['accuracy'])
-
-history = model.fit(train_images, train_labels, epochs=10,
- validation_data=(test_images, test_labels))
+model = models.Sequential()
+model.add(layers.Conv2D(32, (3, 3), activation='relu', input_shape=(32, 32, 3)))
+model.add(layers.MaxPooling2D((2, 2)))
+model.add(layers.Conv2D(64, (3, 3), activation='relu'))
+model.add(layers.MaxPooling2D((2, 2)))
+model.add(layers.Conv2D(64, (3, 3), activation='relu'))
+
+# Let's display the architecture of our model so far.
+
+model.summary()
+
+You can see that the output of every Conv2D and MaxPooling2D layer is a 3D tensor of shape (height, width, channels). The width and height dimensions tend to shrink as you go deeper in the network. The number of output channels for each Conv2D layer is controlled by the first argument (e.g., 32 or 64). Typically, as the width and height shrink, you can afford (computationally) to add more output channels in each Conv2D layer.
+
@@ -432,6 +446,8 @@ history = model
87
88
89
+ 90
+ 91
»
diff --git a/doc/pub/week42/html/._week42-bs088.html b/doc/pub/week42/html/._week42-bs088.html
index 72222eaf2..eb6e406ac 100644
--- a/doc/pub/week42/html/._week42-bs088.html
+++ b/doc/pub/week42/html/._week42-bs088.html
@@ -172,6 +172,7 @@ Automatically generated HTML file from DocOnce source
2,
None,
'convolutional-neural-networks-recognizing-images'),
+ ('What is the Difference', 2, None, 'what-is-the-difference'),
('Neural Networks vs CNNs', 2, None, 'neural-networks-vs-cnns'),
('Why CNNS for images, sound files, medical images from CT scans '
'etc?',
@@ -221,10 +222,8 @@ Automatically generated HTML file from DocOnce source
2,
None,
'final-words-on-fourier-transforms'),
- ('Convolution Examples: Probability Theory',
- 2,
- None,
- 'convolution-examples-probability-theory'),
+ ('Two-dimensional Objects', 2, None, 'two-dimensional-objects'),
+ ('Cross-Correlation', 2, None, 'cross-correlation'),
('More on Dimensionalities', 2, None, 'more-on-dimensionalities'),
('Further Dimensionality Remarks',
2,
@@ -351,43 +350,45 @@ MathJax.Hub.Config({
Solving the wave equation - the full program using Autograd
Resources on differential equations and deep learning
Convolutional Neural Networks (recognizing images)
- Neural Networks vs CNNs
- Why CNNS for images, sound files, medical images from CT scans etc?
- Regular NNs don’t scale well to full images
- 3D volumes of neurons
- Layers used to build CNNs
- Transforming images
- CNNs in brief
- Mathematics of CNNs
- Convolution Examples: Polynomial multiplication
- Efficient Polynomial Multiplication
- A more efficient way of coding the above Convolution
- Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
- Principle of Superposition
- Simple Code Example
- Wrapping up Fourier transforms
- Finding the Coefficients
- Final words on Fourier Transforms
- Convolution Examples: Probability Theory
- More on Dimensionalities
- Further Dimensionality Remarks
- CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
- Setting it up
- The MNIST dataset again
- Strong correlations
- Layers of a CNN
- Systematic reduction
- Prerequisites: Collect and pre-process data
- Importing Keras and Tensorflow
- Running with Keras
- Final part
- Final visualization
- The CIFAR01 data set
- Verifying the data set
- Set up the model
- Add Dense layers on top
- Compile and train the model
- Finally, evaluate the model
+ What is the Difference
+ Neural Networks vs CNNs
+ Why CNNS for images, sound files, medical images from CT scans etc?
+ Regular NNs don’t scale well to full images
+ 3D volumes of neurons
+ Layers used to build CNNs
+ Transforming images
+ CNNs in brief
+ Mathematics of CNNs
+ Convolution Examples: Polynomial multiplication
+ Efficient Polynomial Multiplication
+ A more efficient way of coding the above Convolution
+ Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
+ Principle of Superposition
+ Simple Code Example
+ Wrapping up Fourier transforms
+ Finding the Coefficients
+ Final words on Fourier Transforms
+ Two-dimensional Objects
+ Cross-Correlation
+ More on Dimensionalities
+ Further Dimensionality Remarks
+ CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
+ Setting it up
+ The MNIST dataset again
+ Strong correlations
+ Layers of a CNN
+ Systematic reduction
+ Prerequisites: Collect and pre-process data
+ Importing Keras and Tensorflow
+ Running with Keras
+ Final part
+ Final visualization
+ The CIFAR01 data set
+ Verifying the data set
+ Set up the model
+ Add Dense layers on top
+ Compile and train the model
+ Finally, evaluate the model
@@ -403,24 +404,31 @@ MathJax.Hub.Config({
-Finally, evaluate the model
+Add Dense layers on top
+
+
+To complete our model, you will feed the last output tensor from the
+convolutional base (of shape (4, 4, 64)) into one or more Dense layers
+to perform classification. Dense layers take vectors as input (which
+are 1D), while the current output is a 3D tensor. First, you will
+flatten (or unroll) the 3D output to 1D, then add one or more Dense
+layers on top. CIFAR has 10 output classes, so you use a final Dense
+layer with 10 outputs and a softmax activation.
-
plt.plot(history.history['accuracy'], label='accuracy')
-plt.plot(history.history['val_accuracy'], label = 'val_accuracy')
-plt.xlabel('Epoch')
-plt.ylabel('Accuracy')
-plt.ylim([0.5, 1])
-plt.legend(loc='lower right')
+model.add(layers.Flatten())
+model.add(layers.Dense(64, activation='relu'))
+model.add(layers.Dense(10))
+Here's the complete architecture of our model.
-test_loss, test_acc = model.evaluate(test_images, test_labels, verbose=2)
-
-print(test_acc)
+model.summary()
+As you can see, our (4, 4, 64) outputs were flattened into vectors of shape (1024) before going through two Dense layers.
+
diff --git a/doc/pub/week42/html/week42-bs.html b/doc/pub/week42/html/week42-bs.html
index 0a10af5d5..5522dc854 100644
--- a/doc/pub/week42/html/week42-bs.html
+++ b/doc/pub/week42/html/week42-bs.html
@@ -172,6 +172,7 @@ Automatically generated HTML file from DocOnce source
2,
None,
'convolutional-neural-networks-recognizing-images'),
+ ('What is the Difference', 2, None, 'what-is-the-difference'),
('Neural Networks vs CNNs', 2, None, 'neural-networks-vs-cnns'),
('Why CNNS for images, sound files, medical images from CT scans '
'etc?',
@@ -221,10 +222,8 @@ Automatically generated HTML file from DocOnce source
2,
None,
'final-words-on-fourier-transforms'),
- ('Convolution Examples: Probability Theory',
- 2,
- None,
- 'convolution-examples-probability-theory'),
+ ('Two-dimensional Objects', 2, None, 'two-dimensional-objects'),
+ ('Cross-Correlation', 2, None, 'cross-correlation'),
('More on Dimensionalities', 2, None, 'more-on-dimensionalities'),
('Further Dimensionality Remarks',
2,
@@ -351,43 +350,45 @@ MathJax.Hub.Config({
Solving the wave equation - the full program using Autograd
Resources on differential equations and deep learning
Convolutional Neural Networks (recognizing images)
- Neural Networks vs CNNs
- Why CNNS for images, sound files, medical images from CT scans etc?
- Regular NNs don’t scale well to full images
- 3D volumes of neurons
- Layers used to build CNNs
- Transforming images
- CNNs in brief
- Mathematics of CNNs
- Convolution Examples: Polynomial multiplication
- Efficient Polynomial Multiplication
- A more efficient way of coding the above Convolution
- Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
- Principle of Superposition
- Simple Code Example
- Wrapping up Fourier transforms
- Finding the Coefficients
- Final words on Fourier Transforms
- Convolution Examples: Probability Theory
- More on Dimensionalities
- Further Dimensionality Remarks
- CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
- Setting it up
- The MNIST dataset again
- Strong correlations
- Layers of a CNN
- Systematic reduction
- Prerequisites: Collect and pre-process data
- Importing Keras and Tensorflow
- Running with Keras
- Final part
- Final visualization
- The CIFAR01 data set
- Verifying the data set
- Set up the model
- Add Dense layers on top
- Compile and train the model
- Finally, evaluate the model
+ What is the Difference
+ Neural Networks vs CNNs
+ Why CNNS for images, sound files, medical images from CT scans etc?
+ Regular NNs don’t scale well to full images
+ 3D volumes of neurons
+ Layers used to build CNNs
+ Transforming images
+ CNNs in brief
+ Mathematics of CNNs
+ Convolution Examples: Polynomial multiplication
+ Efficient Polynomial Multiplication
+ A more efficient way of coding the above Convolution
+ Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
+ Principle of Superposition
+ Simple Code Example
+ Wrapping up Fourier transforms
+ Finding the Coefficients
+ Final words on Fourier Transforms
+ Two-dimensional Objects
+ Cross-Correlation
+ More on Dimensionalities
+ Further Dimensionality Remarks
+ CNNs in more detail, building convolutional neural networks in Tensorflow and Keras
+ Setting it up
+ The MNIST dataset again
+ Strong correlations
+ Layers of a CNN
+ Systematic reduction
+ Prerequisites: Collect and pre-process data
+ Importing Keras and Tensorflow
+ Running with Keras
+ Final part
+ Final visualization
+ The CIFAR01 data set
+ Verifying the data set
+ Set up the model
+ Add Dense layers on top
+ Compile and train the model
+ Finally, evaluate the model
@@ -422,7 +423,7 @@ MathJax.Hub.Config({
[2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University
-
Oct 21, 2021
Oct 22, 2021
@@ -446,7 +447,7 @@ MathJax.Hub.Config({
9
10
...
- 89
+ 91
»
diff --git a/doc/pub/week42/html/week42-reveal.html b/doc/pub/week42/html/week42-reveal.html
index 61a89dc27..d8afb546b 100644
--- a/doc/pub/week42/html/week42-reveal.html
+++ b/doc/pub/week42/html/week42-reveal.html
@@ -148,7 +148,7 @@ MathJax.Hub.Config({
[2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University
-
Oct 21, 2021
Oct 22, 2021
@@ -184,6 +184,17 @@ MathJax.Hub.Config({
Excellent lectures on CNNs
+
+
+
+
+And Lecture material on CNNs
+
@@ -2984,9 +2995,14 @@ pixels on one end to class scores at the other. And they still have a
loss function (for example Softmax) on the last (fully-connected) layer
and all the tips/tricks we developed for learning regular Neural
Networks still apply (back propagation, gradient descent etc etc).
+
+
+
+
+What is the Difference
-What is the difference? CNN architectures make the explicit assumption that
+CNN architectures make the explicit assumption that
the inputs are images, which allows us to encode certain properties
into the architecture. These then make the forward function more
efficient to implement and vastly reduce the amount of parameters in
@@ -3610,10 +3626,42 @@ plt.show()
-Convolution Examples: Probability Theory
+Two-dimensional Objects
-More text will be added here
+We often use convolutions over more than one dimension at a time. If
+we have a two-dimensional image \( I \) as input, we can have a filter
+defined by a two-dimensional kernel \( K \). This leads to an output \( S \)
+
+
+$$
+S_(i,j)=(I * K)(i,j) = \sum_m\sum_n I(m,n)K(i-m,j-n).
+$$
+
+
+
+Convolution is a commutatitave process, which means we can rewrite this equation as
+
+$$
+S_(i,j)=(I * K)(i,j) = \sum_m\sum_n I(i-m,j-n)K(m,n).
+$$
+
+
+
+Normally the latter is more straightforward to implement in a machine elarning library since there is less variation in the range of values of \( m \) and \( n \).
+
+
+
+
+Cross-Correlation
+
+
+Many deep learning libraries implement cross-correlation instead of convolution
+
+$$
+S_(i,j)=(I * K)(i,j) = \sum_m\sum_n I(i+m,j-+)K(m,n).
+$$
+
diff --git a/doc/pub/week42/html/week42-solarized.html b/doc/pub/week42/html/week42-solarized.html
index 88ec1448d..662c3389d 100644
--- a/doc/pub/week42/html/week42-solarized.html
+++ b/doc/pub/week42/html/week42-solarized.html
@@ -192,6 +192,7 @@ div { text-align: justify; text-justify: inter-word; }
2,
None,
'convolutional-neural-networks-recognizing-images'),
+ ('What is the Difference', 2, None, 'what-is-the-difference'),
('Neural Networks vs CNNs', 2, None, 'neural-networks-vs-cnns'),
('Why CNNS for images, sound files, medical images from CT scans '
'etc?',
@@ -241,10 +242,8 @@ div { text-align: justify; text-justify: inter-word; }
2,
None,
'final-words-on-fourier-transforms'),
- ('Convolution Examples: Probability Theory',
- 2,
- None,
- 'convolution-examples-probability-theory'),
+ ('Two-dimensional Objects', 2, None, 'two-dimensional-objects'),
+ ('Cross-Correlation', 2, None, 'cross-correlation'),
('More on Dimensionalities', 2, None, 'more-on-dimensionalities'),
('Further Dimensionality Remarks',
2,
@@ -324,7 +323,7 @@ MathJax.Hub.Config({
[2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University
-
Oct 21, 2021
Oct 22, 2021
@@ -359,6 +358,20 @@ MathJax.Hub.Config({
+
+
+
+
+
+And Lecture material on CNNs
+
+
+
@@ -3025,7 +3038,12 @@ and all the tips/tricks we developed for learning regular Neural
Networks still apply (back propagation, gradient descent etc etc).
-What is the difference? CNN architectures make the explicit assumption that
+
+
+What is the Difference
+
+
+CNN architectures make the explicit assumption that
the inputs are images, which allows us to encode certain properties
into the architecture. These then make the forward function more
efficient to implement and vastly reduce the amount of parameters in
@@ -3603,10 +3621,36 @@ plt.show()
-
Convolution Examples: Probability Theory
+Two-dimensional Objects
-More text will be added here
+We often use convolutions over more than one dimension at a time. If
+we have a two-dimensional image \( I \) as input, we can have a filter
+defined by a two-dimensional kernel \( K \). This leads to an output \( S \)
+
+$$
+S_(i,j)=(I * K)(i,j) = \sum_m\sum_n I(m,n)K(i-m,j-n).
+$$
+
+
+Convolution is a commutatitave process, which means we can rewrite this equation as
+$$
+S_(i,j)=(I * K)(i,j) = \sum_m\sum_n I(i-m,j-n)K(m,n).
+$$
+
+
+Normally the latter is more straightforward to implement in a machine elarning library since there is less variation in the range of values of \( m \) and \( n \).
+
+
+
+
+
Cross-Correlation
+
+
+Many deep learning libraries implement cross-correlation instead of convolution
+$$
+S_(i,j)=(I * K)(i,j) = \sum_m\sum_n I(i+m,j-+)K(m,n).
+$$
diff --git a/doc/pub/week42/html/week42.html b/doc/pub/week42/html/week42.html
index ee0a7526c..24f9f17a4 100644
--- a/doc/pub/week42/html/week42.html
+++ b/doc/pub/week42/html/week42.html
@@ -197,6 +197,7 @@ div { text-align: justify; text-justify: inter-word; }
2,
None,
'convolutional-neural-networks-recognizing-images'),
+ ('What is the Difference', 2, None, 'what-is-the-difference'),
('Neural Networks vs CNNs', 2, None, 'neural-networks-vs-cnns'),
('Why CNNS for images, sound files, medical images from CT scans '
'etc?',
@@ -246,10 +247,8 @@ div { text-align: justify; text-justify: inter-word; }
2,
None,
'final-words-on-fourier-transforms'),
- ('Convolution Examples: Probability Theory',
- 2,
- None,
- 'convolution-examples-probability-theory'),
+ ('Two-dimensional Objects', 2, None, 'two-dimensional-objects'),
+ ('Cross-Correlation', 2, None, 'cross-correlation'),
('More on Dimensionalities', 2, None, 'more-on-dimensionalities'),
('Further Dimensionality Remarks',
2,
@@ -329,7 +328,7 @@ MathJax.Hub.Config({
[2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University
-
Oct 21, 2021
Oct 22, 2021
@@ -364,6 +363,20 @@ MathJax.Hub.Config({
+
+
+
+
+
+And Lecture material on CNNs
+
+
+
@@ -3030,7 +3043,12 @@ and all the tips/tricks we developed for learning regular Neural
Networks still apply (back propagation, gradient descent etc etc).
-What is the difference? CNN architectures make the explicit assumption that
+
+
+What is the Difference
+
+
+CNN architectures make the explicit assumption that
the inputs are images, which allows us to encode certain properties
into the architecture. These then make the forward function more
efficient to implement and vastly reduce the amount of parameters in
@@ -3608,10 +3626,36 @@ plt.show()
-
Convolution Examples: Probability Theory
+Two-dimensional Objects
-More text will be added here
+We often use convolutions over more than one dimension at a time. If
+we have a two-dimensional image \( I \) as input, we can have a filter
+defined by a two-dimensional kernel \( K \). This leads to an output \( S \)
+
+$$
+S_(i,j)=(I * K)(i,j) = \sum_m\sum_n I(m,n)K(i-m,j-n).
+$$
+
+
+Convolution is a commutatitave process, which means we can rewrite this equation as
+$$
+S_(i,j)=(I * K)(i,j) = \sum_m\sum_n I(i-m,j-n)K(m,n).
+$$
+
+
+Normally the latter is more straightforward to implement in a machine elarning library since there is less variation in the range of values of \( m \) and \( n \).
+
+
+
+
+
Cross-Correlation
+
+
+Many deep learning libraries implement cross-correlation instead of convolution
+$$
+S_(i,j)=(I * K)(i,j) = \sum_m\sum_n I(i+m,j-+)K(m,n).
+$$
diff --git a/doc/pub/week42/ipynb/ipynb-week42-src.tar.gz b/doc/pub/week42/ipynb/ipynb-week42-src.tar.gz
index d496685a2..59a667ebd 100644
Binary files a/doc/pub/week42/ipynb/ipynb-week42-src.tar.gz and b/doc/pub/week42/ipynb/ipynb-week42-src.tar.gz differ
diff --git a/doc/pub/week42/ipynb/week42.ipynb b/doc/pub/week42/ipynb/week42.ipynb
index 7834c4793..3a42f7616 100644
--- a/doc/pub/week42/ipynb/week42.ipynb
+++ b/doc/pub/week42/ipynb/week42.ipynb
@@ -10,7 +10,7 @@
" \n",
"**Morten Hjorth-Jensen**, Department of Physics, University of Oslo and Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University\n",
"\n",
- "Date: **Oct 21, 2021**\n",
+ "Date: **Oct 22, 2021**\n",
"\n",
"Copyright 1999-2021, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license\n",
"\n",
@@ -39,6 +39,19 @@
"\n",
"* [Video on Convolutional Neural Networks from MIT](https://www.youtube.com/watch?v=iaSUYvmCekI&ab_channel=AlexanderAmini)\n",
"\n",
+ "* [Video on CNNs from Stanford](https://www.youtube.com/watch?v=bNb2fEVKeEo&list=PLC1qU-LWwrF64f4QKQT-Vg5Wr4qEE1Zxk&index=6&ab_channel=StanfordUniversitySchoolofEngineering)\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "**And Lecture material on CNNs.**\n",
+ "\n",
+ "* [Lectures from IN5400 spring 2019](https://www.uio.no/studier/emner/matnat/ifi/IN5400/v19/material/week5/in5400_2019_week5_convolutional_nerual_networks.pdf)\n",
+ "\n",
+ "* [Lectures from IN5400 spring 2021](https://www.uio.no/studier/emner/matnat/ifi/IN5400/v21/lecture-slides/in5400_2021_w5_lecture_convolutions.pdf)\n",
+ "\n",
+ "* [See also Michael Nielsen's Lectures](http://neuralnetworksanddeeplearning.com/chap6.html)\n",
+ "\n",
"\n",
"\n",
"\n",
@@ -3121,7 +3134,9 @@
"and all the tips/tricks we developed for learning regular Neural\n",
"Networks still apply (back propagation, gradient descent etc etc).\n",
"\n",
- "What is the difference? **CNN architectures make the explicit assumption that\n",
+ "## What is the Difference\n",
+ "\n",
+ "**CNN architectures make the explicit assumption that\n",
"the inputs are images, which allows us to encode certain properties\n",
"into the architecture. These then make the forward function more\n",
"efficient to implement and vastly reduce the amount of parameters in\n",
@@ -3876,11 +3891,64 @@
"cell_type": "markdown",
"metadata": {},
"source": [
- "## Convolution Examples: Probability Theory\n",
+ "## Two-dimensional Objects\n",
"\n",
- "More text will be added here\n",
+ "We often use convolutions over more than one dimension at a time. If\n",
+ "we have a two-dimensional image $I$ as input, we can have a **filter**\n",
+ "defined by a two-dimensional **kernel** $K$. This leads to an output $S$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "S_(i,j)=(I * K)(i,j) = \\sum_m\\sum_n I(m,n)K(i-m,j-n).\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Convolution is a commutatitave process, which means we can rewrite this equation as"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "S_(i,j)=(I * K)(i,j) = \\sum_m\\sum_n I(i-m,j-n)K(m,n).\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Normally the latter is more straightforward to implement in a machine elarning library since there is less variation in the range of values of $m$ and $n$.\n",
+ "\n",
+ "## Cross-Correlation\n",
"\n",
"\n",
+ "\n",
+ "Many deep learning libraries implement cross-correlation instead of convolution"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "S_(i,j)=(I * K)(i,j) = \\sum_m\\sum_n I(i+m,j-+)K(m,n).\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
"## More on Dimensionalities\n",
"\n",
"In feilds like signal processing (and imaging as well), one designs\n",
diff --git a/doc/src/week42/week42.do.txt b/doc/src/week42/week42.do.txt
index 33a53b559..cf2e9975c 100644
--- a/doc/src/week42/week42.do.txt
+++ b/doc/src/week42/week42.do.txt
@@ -20,6 +20,14 @@ DATE: today
!bblock Excellent lectures on CNNs
* "Video on Convolutional Neural Networks from MIT":"https://www.youtube.com/watch?v=iaSUYvmCekI&ab_channel=AlexanderAmini"
+* "Video on CNNs from Stanford":"https://www.youtube.com/watch?v=bNb2fEVKeEo&list=PLC1qU-LWwrF64f4QKQT-Vg5Wr4qEE1Zxk&index=6&ab_channel=StanfordUniversitySchoolofEngineering"
+!eblock
+
+
+!bblock And Lecture material on CNNs
+* "Lectures from IN5400 spring 2019":"https://www.uio.no/studier/emner/matnat/ifi/IN5400/v19/material/week5/in5400_2019_week5_convolutional_nerual_networks.pdf"
+* "Lectures from IN5400 spring 2021":"https://www.uio.no/studier/emner/matnat/ifi/IN5400/v21/lecture-slides/in5400_2021_w5_lecture_convolutions.pdf"
+* "See also Michael Nielsen's Lectures":"http://neuralnetworksanddeeplearning.com/chap6.html"
!eblock
@@ -2444,7 +2452,10 @@ loss function (for example Softmax) on the last (fully-connected) layer
and all the tips/tricks we developed for learning regular Neural
Networks still apply (back propagation, gradient descent etc etc).
-What is the difference? _CNN architectures make the explicit assumption that
+!split
+===== What is the Difference =====
+
+_CNN architectures make the explicit assumption that
the inputs are images, which allows us to encode certain properties
into the architecture. These then make the forward function more
efficient to implement and vastly reduce the amount of parameters in
@@ -2957,9 +2968,38 @@ plt.show()
!split
-===== Convolution Examples: Probability Theory =====
+===== Two-dimensional Objects =====
-More text will be added here
+We often use convolutions over more than one dimension at a time. If
+we have a two-dimensional image $I$ as input, we can have a _filter_
+defined by a two-dimensional _kernel_ $K$. This leads to an output $S$
+
+!bt
+\[
+S_(i,j)=(I * K)(i,j) = \sum_m\sum_n I(m,n)K(i-m,j-n).
+\]
+!et
+
+Convolution is a commutatitave process, which means we can rewrite this equation as
+!bt
+\[
+S_(i,j)=(I * K)(i,j) = \sum_m\sum_n I(i-m,j-n)K(m,n).
+\]
+!et
+
+Normally the latter is more straightforward to implement in a machine elarning library since there is less variation in the range of values of $m$ and $n$.
+
+!split
+===== Cross-Correlation =====
+
+
+
+Many deep learning libraries implement cross-correlation instead of convolution
+!bt
+\[
+S_(i,j)=(I * K)(i,j) = \sum_m\sum_n I(i+m,j-+)K(m,n).
+\]
+!et
!split