Figure 1: A regular 3-layer Neural Network.
# # ## 3D volumes of neurons # # Convolutional Neural Networks take advantage of the fact that the # input consists of images and they constrain the architecture in a more # sensible way. # # In particular, unlike a regular Neural Network, the # layers of a CNN have neurons arranged in 3 dimensions: width, # height, depth. (Note that the word depth here refers to the third # dimension of an activation volume, not to the depth of a full Neural # Network, which can refer to the total number of layers in a network.) # # To understand it better, the above example of an image # with an input volume of # activations has dimensions $32\times 32\times 3$ (width, height, # depth respectively). # # The neurons in a layer will # only be connected to a small region of the layer before it, instead of # all of the neurons in a fully-connected manner. Moreover, the final # output layer could for this specific image have dimensions $1\times 1 \times 10$, # because by the # end of the CNN architecture we will reduce the full image into a # single vector of class scores, arranged along the depth # dimension. # # # # #
Figure 1: A CNN arranges its neurons in three dimensions (width, height, depth), as visualized in one of the layers. Every layer of a CNN transforms the 3D input volume to a 3D output volume of neuron activations. In this example, the red input layer holds the image, so its width and height would be the dimensions of the image, and the depth would be 3 (Red, Green, Blue channels).
# # ## Layers used to build CNNs # # A simple CNN is a sequence of layers, and every layer of a CNN # transforms one volume of activations to another through a # differentiable function. We use three main types of layers to build # CNN architectures: Convolutional Layer, Pooling Layer, and # Fully-Connected Layer (exactly as seen in regular Neural Networks). We # will stack these layers to form a full CNN architecture. # # A simple CNN for image classification could have the architecture: # # * **INPUT** ($32\times 32 \times 3$) will hold the raw pixel values of the image, in this case an image of width 32, height 32, and with three color channels R,G,B. # # * **CONV** (convolutional )layer will compute the output of neurons that are connected to local regions in the input, each computing a dot product between their weights and a small region they are connected to in the input volume. This may result in volume such as $[32\times 32\times 12]$ if we decided to use 12 filters. # # * **RELU** layer will apply an elementwise activation function, such as the $max(0,x)$ thresholding at zero. This leaves the size of the volume unchanged ($[32\times 32\times 12]$). # # * **POOL** (pooling) layer will perform a downsampling operation along the spatial dimensions (width, height), resulting in volume such as $[16\times 16\times 12]$. # # * **FC** (i.e. fully-connected) layer will compute the class scores, resulting in volume of size $[1\times 1\times 10]$, where each of the 10 numbers correspond to a class score, such as among the 10 categories of the MNIST images we considered above . As with ordinary Neural Networks and as the name implies, each neuron in this layer will be connected to all the numbers in the previous volume. # ## Transforming images # # CNNs transform the original image layer by layer from the original # pixel values to the final class scores. # # Observe that some layers contain # parameters and other don’t. In particular, the CNN layers perform # transformations that are a function of not only the activations in the # input volume, but also of the parameters (the weights and biases of # the neurons). On the other hand, the RELU/POOL layers will implement a # fixed function. The parameters in the CONV/FC layers will be trained # with gradient descent so that the class scores that the CNN computes # are consistent with the labels in the training set for each image. # ## CNNs in brief # # In summary: # # * A CNN architecture is in the simplest case a list of Layers that transform the image volume into an output volume (e.g. holding the class scores) # # * There are a few distinct types of Layers (e.g. CONV/FC/RELU/POOL are by far the most popular) # # * Each Layer accepts an input 3D volume and transforms it to an output 3D volume through a differentiable function # # * Each Layer may or may not have parameters (e.g. CONV/FC do, RELU/POOL don’t) # # * Each Layer may or may not have additional hyperparameters (e.g. CONV/FC/POOL do, RELU doesn’t) # # For more material on convolutional networks, we strongly recommend # the course # [CS231](http://cs231n.github.io/convolutional-networks/) which is taught at Stanford University (consistently ranked as one of the top computer science programs in the world). [Michael Nielsen's book is a must read, in particular chapter 6 which deals with CNNs](http://neuralnetworksanddeeplearning.com/chap6.html). # # The textbook by Goodfellow et al, see chapter 9 contains an in depth discussion as well. # ## Key Idea # # A dense neural network is representd by an affine operation (like matrix-matrix multiplication) where all parameters are included. # # The key idea in CNNs for say imaging is that in images neighbor pixels tend to be related! So we connect # only neighboring neurons in the input instead of connecting all with the first hidden layer. # # We say we perform a filtering (convolution is the mathematical operation). # ## Mathematics of CNNs # # The mathematics of CNNs is based on the mathematical operation of # **convolution**. In mathematics (in particular in functional analysis), # convolution is represented by mathematical 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 # $$ # y(t) = \int x(a) w(t-a) da, # $$ # 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 # $$ # y(t) = \left(x * w\right)(t). # $$ # 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. # ## Convolution Examples: Polynomial multiplication # # 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: # $$ # p(t) = \alpha_0+\alpha_1 t+\alpha_2 t^2, # $$ # and # $$ # s(t) = \beta_0+\beta_1 t+\beta_2 t^2+\beta_3 t^3. # $$ # The polynomial multiplication gives us a new polynomial of degree $5$ # $$ # z(t) = \delta_0+\delta_1 t+\delta_2 t^2+\delta_3 t^3+\delta_4 t^4+\delta_5 t^5. # $$ # ## Efficient Polynomial 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 # $$ # \begin{split} # \delta_0=&\alpha_0\beta_0\\ # \delta_1=&\alpha_1\beta_0+\alpha_1\beta_0\\ # \delta_2=&\alpha_0\beta_2+\alpha_1\beta_1+\alpha_2\beta_0\\ # \delta_3=&\alpha_1\beta_2+\alpha_2\beta_1+\alpha_0\beta_3\\ # \delta_4=&\alpha_2\beta_2+\alpha_1\beta_3\\ # \delta_5=&\alpha_2\beta_3.\\ # \end{split} # $$ # 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 # $$ # \delta_j = \sum_{i=-\infty}^{i=\infty}\alpha_i\beta_{j-i}=(\alpha * \beta)_j, # $$ # or as a double sum with restriction $l=i+j$ # $$ # \delta_l = \sum_{ij}\alpha_i\beta_{j}. # $$ # Do you see a potential drawback with these equations? # ## A more efficient way of coding the above Convolution # # 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 # $$ # \boldsymbol{\delta}=\begin{bmatrix}\alpha_0 & 0 & 0 & 0 \\ # \alpha_1 & \alpha_0 & 0 & 0 \\ # \alpha_2 & \alpha_1 & \alpha_0 & 0 \\ # 0 & \alpha_2 & \alpha_1 & \alpha_0 \\ # 0 & 0 & \alpha_2 & \alpha_1 \\ # 0 & 0 & 0 & \alpha_2 # \end{bmatrix}\begin{bmatrix} \beta_0 \\ \beta_1 \\ \beta_2 \\ \beta_3\end{bmatrix}. # $$ # 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 # $$ # \boldsymbol{\delta}=\begin{bmatrix}\beta_0 & 0 & 0 \\ # \beta_1 & \beta_0 & 0 \\ # \beta_2 & \beta_1 & \beta_0 \\ # \beta_3 & \beta_2 & \beta_1 \\ # 0 & \beta_3 & \beta_2 \\ # 0 & 0 & \beta_3 # \end{bmatrix}\begin{bmatrix} \alpha_0 \\ \alpha_1 \\ \alpha_2\end{bmatrix}. # $$ # 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. # # Does the number of floating point operations change here when we use the commutative property? # # The above matrices are examples of so-called [Toeplitz # matrices](https://link.springer.com/book/10.1007/978-93-86279-04-0). A # Toeplitz matrix is a matrix in which each descending diagonal from # left to right is constant. For instance the last matrix, which we # rewrite as # $$ # \boldsymbol{A}=\begin{bmatrix}a_0 & 0 & 0 \\ # a_1 & a_0 & 0 \\ # a_2 & a_1 & a_0 \\ # a_3 & a_2 & a_1 \\ # 0 & a_3 & a_2 \\ # 0 & 0 & a_3 # \end{bmatrix}, # $$ # with elements $a_{ii}=a_{i+1,j+1}=a_{i-j}$ is an example of a Toeplitz # matrix. Such a matrix does not need to be a square matrix. Toeplitz # matrices are also closely connected with Fourier series discussed # below, because the multiplication operator by a trigonometric # polynomial, compressed to a finite-dimensional space, can be # represented by such a matrix. The example above shows that we can # represent linear convolution as multiplication of a Toeplitz matrix by # a vector. # ## Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms) # # For problems with so-called harmonic oscillations, given by for example the following differential equation # $$ # m\frac{d^2x}{dt^2}+\eta\frac{dx}{dt}+x(t)=F(t), # $$ # 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. # # If one has several driving forces, $F(t)=\sum_n F_n(t)$, one can find # the particular solution $x_{pn}(t)$ to the above differential equation for each $F_n$. The particular # solution for the entire driving force is then given by a series like # # # # $$ # \begin{equation} # x_p(t)=\sum_nx_{pn}(t). # \label{_auto1} \tag{1} # \end{equation} # $$ # This is known as the principle of superposition. It only applies when # the homogenous equation is linear. # 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 $t$ our function repeats itself periodically after a period $\tau$, that is # $$ # \begin{eqnarray} # F(t+\tau)=F(t). # \end{eqnarray} # $$ # One example of a non-sinusoidal periodic force is a square wave. Many # components in electric circuits are non-linear, for example diodes. This # makes many wave forms non-sinusoidal even when the circuits are being # driven by purely sinusoidal sources. # ## Simple Code Example # # 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 $\tau=0.2$. # In[1]: get_ipython().run_line_magic('matplotlib', 'inline') 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, # # # # $$ # \begin{equation} # F(t)=\frac{f_0}{2}+\sum_{n>0} f_n\cos(2n\pi t/\tau)+g_n\sin(2n\pi t/\tau). # \label{_auto2} \tag{2} # \end{equation} # $$ # ## Wrapping up Fourier transforms # # 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 $\omega\equiv # 2\pi/\tau$, # # # # $$ # \begin{equation} # \label{eq:fourierdef1} \tag{3} # F(t)=\frac{f_0}{2}+\sum_{n>0}f_n\cos(n\omega t)+g_n\sin(n\omega t). # \end{equation} # $$ # The solutions for $x(t)$ then come from replacing $\omega$ with # $n\omega$ for each term in the particular solution, # $$ # \begin{eqnarray} # x_p(t)&=&\frac{f_0}{2k}+\sum_{n>0} \alpha_n\cos(n\omega t-\delta_n)+\beta_n\sin(n\omega t-\delta_n),\\ # \nonumber # \alpha_n&=&\frac{f_n/m}{\sqrt{((n\omega)^2-\omega_0^2)+4\beta^2n^2\omega^2}},\\ # \nonumber # \beta_n&=&\frac{g_n/m}{\sqrt{((n\omega)^2-\omega_0^2)+4\beta^2n^2\omega^2}},\\ # \nonumber # \delta_n&=&\tan^{-1}\left(\frac{2\beta n\omega}{\omega_0^2-n^2\omega^2}\right). # \end{eqnarray} # $$ # ## Finding the Coefficients # # Because the forces have been applied for a long time, any non-zero # damping eliminates the homogenous parts of the solution. We need then # 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 # # # # $$ # \begin{eqnarray} # \label{eq:fourierdef2} \tag{4} # f_n&=&\frac{2}{\tau}\int_{-\tau/2}^{\tau/2} dt~F(t)\cos(2n\pi t/\tau),\\ # \nonumber # g_n&=&\frac{2}{\tau}\int_{-\tau/2}^{\tau/2} dt~F(t)\sin(2n\pi t/\tau). # \end{eqnarray} # $$ # To check the consistency of these expressions and to verify # Eq. ([4](#eq:fourierdef2)), one can insert the expansion of $F(t)$ in # Eq. ([3](#eq:fourierdef1)) into the expression for the coefficients in # Eq. ([4](#eq:fourierdef2)) and see whether # $$ # f_n=\frac{2}{\tau}\int_{-\tau/2}^{\tau/2} dt~\left\{\frac{f_0}{2}+\sum_{m>0}f_m\cos(m\omega t)+g_m\sin(m\omega t)\right\}\cos(n\omega t). # $$ # 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 # # # # $$ # \begin{equation} # \int_{-\tau/2}^{\tau/2}dt~\cos^2(m\omega t)=\frac{\tau}{2}, # \label{_auto3} \tag{5} # \end{equation} # $$ # and # $$ # f_n=\frac{2}{\tau}\int_{-\tau/2}^{\tau/2} dt~f_n/2=f_n. # $$ # The same method can be used to check for the consistency of $g_n$. # ## Final words on Fourier Transforms # # 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. # In[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 =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() # ### Fourier transforms and convolution # # We can use Fourier transforms in our studies of convolution as well. To see this, assume we have two functions $f$ and $g$ and their corresponding Fourier transforms $\hat{f}$ and $\hat{g}$. We remind the reader that the Fourier transform reads (say for the function $f$) # $$ # \hat{f}(y)=\boldsymbol{F}[f(y)]=\frac{1}{2\pi}\int_{-\infty}^{\infty} d\omega \exp{-i\omega y} f(\omega), # $$ # and similarly we have # $$ # \hat{g}(y)=\boldsymbol{F}[g(y)]=\frac{1}{2\pi}\int_{-\infty}^{\infty} d\omega \exp{-i\omega y} g(\omega). # $$ # The inverse Fourier transform is given by # $$ # \boldsymbol{F}^{-1}[g(y)]=\frac{1}{2\pi}\int_{-\infty}^{\infty} d\omega \exp{i\omega y} g(\omega). # $$ # The inverse Fourier transform of the product of the two functions $\hat{f}\hat{g}$ can be written as # $$ # \boldsymbol{F}^{-1}[(\hat{f}\hat{g})(x)]=\frac{1}{2\pi}\int_{-\infty}^{\infty} d\omega \exp{i\omega x} \hat{f}(\omega)\hat{g}(\omega). # $$ # We can rewrite the latter as # $$ # \boldsymbol{F}^{-1}[(\hat{f}\hat{g})(x)]=\int_{-\infty}^{\infty} d\omega \exp{i\omega x} \hat{f}(\omega)\left[\frac{1}{2\pi}\int_{-\infty}^{\infty}g(y)dy \exp{-i\omega y}\right]=\frac{1}{2\pi}\int_{-\infty}^{\infty}dy g(y)\int_{-\infty}^{\infty} d\omega \hat{f}(\omega) \exp{i\omega(x- y)}, # $$ # which is simply # $$ # \boldsymbol{F}^{-1}[(\hat{f}\hat{g})(x)]=\int_{-\infty}^{\infty}dy g(y)f(x-y)=(f*g)(x), # $$ # the convolution of the functions $f$ and $g$. # ## Two-dimensional Objects # # We are now ready to start studying the discrete convolutions relevant for convolutional neural networks. # 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 larning library since there is less variation in the range of values of $m$ and $n$. # # Many deep learning libraries implement cross-correlation instead of convolution (although it is referred to s convolution) # $$ # S_(i,j)=(I * K)(i,j) = \sum_m\sum_n I(i+m,j+n)K(m,n). # $$ # ## More on Dimensionalities # # In fields 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 # $$ # \mathrm{NumberParameters}=10^{10}+10^4+10^4+1 \approx 10^{10}, # $$ # that is ten billion parameters to determine. # ## Further Dimensionality Remarks # # 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. # ## CNNs in more detail # # Let assume we have an input matrix $I$ of dimensionality $3\times 3$ # and a $2\times 2$ filter $W$ given by the following matrices # $$ # \boldsymbol{I}=\begin{bmatrix}i_{00} & i_{01} & i_{02} \\ # i_{10} & i_{11} & i_{12} \\ # i_{20} & i_{21} & i_{22} \end{bmatrix}, # $$ # and # $$ # \boldsymbol{W}=\begin{bmatrix}w_{00} & w_{01} \\ # w_{10} & w_{11}\end{bmatrix}. # $$ # We introduce now the hyperparameter $S$ **stride**. Stride represents how the filter $W$ moves the convolution process on the matrix $I$. # We strongly recommend the repository on [Arithmetic of deep learning by Dumoulin and Visin](https://github.com/vdumoulin/conv_arithmetic) # # Here we set the stride equal to $S=1$, which means that, starting with the element $i_{00}$, the filter will act on $2\times 2$ submatrices each time, starting with the upper corner and moving according to the stride value column by column. # # Here we perform the operation # $$ # S_(i,j)=(I * W)(i,j) = \sum_m\sum_n I(i-m,j-n)W(m,n), # $$ # and obtain # $$ # \boldsymbol{S}=\begin{bmatrix}i_{00}w_{00}+i_{01}w_{01}+i_{10}w_{10}+i_{11}w_{11} & i_{01}w_{00}+i_{02}w_{01}+i_{11}w_{10}+i_{12}w_{11} \\ # i_{10}w_{00}+i_{11}w_{01}+i_{20}w_{10}+i_{21}w_{11} & i_{11}w_{00}+i_{12}w_{01}+i_{21}w_{10}+i_{22}w_{11}\end{bmatrix}. # $$ # We can rewrite this operation in terms of a matrix-vector multiplication by defining a new vector where we flatten out the inputs as a vector $\boldsymbol{I}'$ of length $9$ and # a matrix $\boldsymbol{W}'$ with dimension $4\times 9$ as # $$ # \boldsymbol{I}'=\begin{bmatrix}i_{00} \\ i_{01} \\ i_{02} \\ i_{10} \\ i_{11} \\ i_{12} \\ i_{20} \\ i_{21} \\ i_{22} \end{bmatrix}, # $$ # and the new matrix # $$ # \boldsymbol{W}'=\begin{bmatrix} w_{00} & w_{01} & 0 & w_{10} & w_{11} & 0 & 0 & 0 & 0 \\ # 0 & w_{00} & w_{01} & 0 & w_{10} & w_{11} & 0 & 0 & 0 \\ # 0 & 0 & 0 & w_{00} & w_{01} & 0 & w_{10} & w_{11} & 0 \\ # 0 & 0 & 0 & 0 & w_{00} & w_{01} & 0 & w_{10} & w_{11}\end{bmatrix}. # $$ # We see easily that performing the matrix-vector multiplication $\boldsymbol{W}'\boldsymbol{I}'$ is the same as the above convolution with stride $S=1$, that is # $$ # S=(\boldsymbol{W}*\boldsymbol{I}), # $$ # is now given by $\boldsymbol{W}'\boldsymbol{I}'$ which is a vector of length $4$ instead of the originally resulting $2\times 2$ output matrix. # # The collection of kernels/filters $W$ defining a discrete convolution has a shape # corresponding to some permutation of $(n, m, k_1, \ldots, k_N)$, where # $$ # \begin{split} # n &\equiv \text{number of output feature maps},\\ # m &\equiv \text{number of input feature maps},\\ # k_j &\equiv \text{kernel size along axis $j$}. # \end{split} # $$ # The following properties affect the output size $o_j$ of a convolutional layer # along axis $j$: # # 1. $i_j$: input size along axis $j$, # # 2. $k_j$: kernel/filter size along axis $j$, # # 3. stride (distance between two consecutive positions of the kernel/filter) along axis $j$, # # 4. zero padding (number of zeros concatenated at the beginning and at the end of an axis) along axis $j$. # # For instance, the above examples shows a $2\times 2$ kernel/filter $\boldsymbol{W}$ applied to a $3 \times 3$ input padded with a $0 \times 0$ # border of zeros using $1 \times 1$ strides. # # Note that strides constitute a form of **subsampling**. As an alternative to # being interpreted as a measure of how much the kernel/filter is translated, strides # can also be viewed as how much of the output is retained. For instance, moving # the kernel by hops of two is equivalent to moving the kernel by hops of one but # retaining only odd output elements. # ## Pooling # # In addition to discrete convolutions themselves, {\em pooling\/} operations # make up another important building block in CNNs. Pooling operations reduce # the size of feature maps by using some function to summarize subregions, such # as taking the average or the maximum value. # # Pooling works by sliding a window across the input and feeding the content of # the window to a {\em pooling function}. In some sense, pooling works very much # like a discrete convolution, but replaces the linear combination described by # the kernel with some other function. Poolin # provides an example for average pooling, and # does the same for max pooling. # # The following properties affect the output size $o_j$ of a pooling layer # along axis $j$: # # 1. $i_j$: input size along axis $j$, # # 2. $k_j$: pooling window size along axis $j$, # # 3. $s_j$: stride (distance between two consecutive positions of the pooling window) along axis $j$. # # The analysis of the relationship between convolutional layer properties is eased # by the fact that they don't interact across axes, i.e., the choice of kernel # size, stride and zero padding along axis $j$ only affects the output size of # axis $j$. Because of that, we will focus on the following simplified # setting: # # 1. 2-D discrete convolutions ($N = 2$), # # 2. square inputs ($i_1 = i_2 = i$), # # 3. square kernel size ($k_1 = k_2 = k$), # # 4. same strides along both axes ($s_1 = s_2 = s$), # # 5. same zero padding along both axes ($p_1 = p_2 = p$). # # This facilitates the analysis and the visualization, but keep in mind that the # results outlined here also generalize to the N-D and non-square cases. # ## No zero padding, unit strides # # The simplest case to analyze is when the kernel just slides across every # position of the input (i.e., $s = 1$ and $p = 0$). # # For any $i$ and $k$, and for $s = 1$ and $p = 0$, # $$ # o = (i - k) + 1. # $$ # ## Zero padding, unit strides # # To factor in zero padding (i.e., only restricting to $s = 1$), let's consider # its effect on the effective input size: padding with $p$ zeros changes the # effective input size from $i$ to $i + 2p$. In the general case, we can infer the following # relationship # # For any $i$, $k$ and $p$, and for $s = 1$, # $$ # o = (i - k) + 2p + 1. # $$ # ## Half (same) padding # # Having the output size be the same as the input size (i.e., $o = i$) can be a # desirable property: # # For any $i$ and for $k$ odd ($k = 2n + 1, \quad n \in \mathbb{N}$), $s = 1$ and # $p = \lfloor k / 2 \rfloor = n$, # $$ # \begin{split} # o &= i + 2 \lfloor k / 2 \rfloor - (k - 1) \\ # &= i + 2n - 2n \\ # &= i. # \end{split} # $$ # ## Full padding # # While convolving a kernel generally decreases the output size with # respect to the input size, sometimes the opposite is required. This can be # achieved with proper zero padding: # # For any $i$ and $k$, and for $p = k - 1$ and $s = 1$, # $$ # \begin{split} # o &= i + 2(k - 1) - (k - 1) \\ # &= i + (k - 1). # \end{split} # $$ # This is sometimes referred to as full padding, because in this # setting every possible partial or complete superimposition of the kernel on the # input feature map is taken into account. # ## Pooling arithmetic # # In a neural network, pooling layers provide invariance to small translations of # the input. The most common kind of pooling is **max pooling**, which # consists in splitting the input in (usually non-overlapping) patches and # outputting the maximum value of each patch. Other kinds of pooling exist, e.g., # mean or average pooling, which all share the same idea of aggregating the input # locally by applying a non-linearity to the content of some patches. # # Since pooling does not involve # zero padding, the relationship describing the general case is as follows: # # For any $i$, $k$ and $s$, # $$ # o = \left\lfloor \frac{i - k}{s} \right\rfloor + 1. # $$ # ## CNNs in more detail, building convolutional neural networks in Tensorflow and Keras # # As discussed above, CNNs are neural networks built from the assumption that the inputs # to the network are 2D images. This is important because the number of features or pixels in images # grows very fast with the image size, and an enormous number of weights and biases are needed in order to build an accurate network. # # As before, we still have our input, a hidden layer and an output. What's novel about convolutional networks # are the **convolutional** and **pooling** layers stacked in pairs between the input and the hidden layer. # In addition, the data is no longer represented as a 2D feature matrix, instead each input is a number of 2D # matrices, typically 1 for each color dimension (Red, Green, Blue). # ## Setting it up # # It means that to represent the entire # dataset of images, we require a 4D matrix or **tensor**. This tensor has the dimensions: # $$ # (n_{inputs},\, n_{pixels, width},\, n_{pixels, height},\, depth) . # $$ # ## The MNIST dataset again # # The MNIST dataset consists of grayscale images with a pixel size of # $28\times 28$, meaning we require $28 \times 28 = 724$ weights to each # neuron in the first hidden layer. # # If we were to analyze images of size $128\times 128$ we would require # $128 \times 128 = 16384$ weights to each neuron. Even worse if we were # dealing with color images, as most images are, we have an image matrix # of size $128\times 128$ for each color dimension (Red, Green, Blue), # meaning 3 times the number of weights $= 49152$ are required for every # single neuron in the first hidden layer. # ## Strong correlations # # Images typically have strong local correlations, meaning that a small # part of the image varies little from its neighboring regions. If for # example we have an image of a blue car, we can roughly assume that a # small blue part of the image is surrounded by other blue regions. # # Therefore, instead of connecting every single pixel to a neuron in the # first hidden layer, as we have previously done with deep neural # networks, we can instead connect each neuron to a small part of the # image (in all 3 RGB depth dimensions). The size of each small area is # fixed, and known as a [receptive](https://en.wikipedia.org/wiki/Receptive_field). # ## Layers of a CNN # The layers of a convolutional neural network arrange neurons in 3D: width, height and depth. # The input image is typically a square matrix of depth 3. # # A **convolution** is performed on the image which outputs # a 3D volume of neurons. The weights to the input are arranged in a number of 2D matrices, known as **filters**. # # Each filter slides along the input image, taking the dot product # between each small part of the image and the filter, in all depth # dimensions. This is then passed through a non-linear function, # typically the **Rectified Linear (ReLu)** function, which serves as the # activation of the neurons in the first convolutional layer. This is # further passed through a **pooling layer**, which reduces the size of the # convolutional layer, e.g. by taking the maximum or average across some # small regions, and this serves as input to the next convolutional # layer. # ## Systematic reduction # # By systematically reducing the size of the input volume, through # convolution and pooling, the network should create representations of # small parts of the input, and then from them assemble representations # of larger areas. The final pooling layer is flattened to serve as # input to a hidden layer, such that each neuron in the final pooling # layer is connected to every single neuron in the hidden layer. This # then serves as input to the output layer, e.g. a softmax output for # classification. # ## Prerequisites: Collect and pre-process data # In[3]: # import necessary packages import numpy as np import matplotlib.pyplot as plt from sklearn import datasets # ensure the same random numbers appear every time np.random.seed(0) # display images in notebook get_ipython().run_line_magic('matplotlib', 'inline') plt.rcParams['figure.figsize'] = (12,12) # download MNIST dataset digits = datasets.load_digits() # define inputs and labels inputs = digits.images labels = digits.target # RGB images have a depth of 3 # our images are grayscale so they should have a depth of 1 inputs = inputs[:,:,:,np.newaxis] print("inputs = (n_inputs, pixel_width, pixel_height, depth) = " + str(inputs.shape)) print("labels = (n_inputs) = " + str(labels.shape)) # choose some random images to display n_inputs = len(inputs) indices = np.arange(n_inputs) random_indices = np.random.choice(indices, size=5) for i, image in enumerate(digits.images[random_indices]): plt.subplot(1, 5, i+1) plt.axis('off') plt.imshow(image, cmap=plt.cm.gray_r, interpolation='nearest') plt.title("Label: %d" % digits.target[random_indices[i]]) plt.show() # ## Importing Keras and Tensorflow # In[4]: 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) # ## Running with Keras # In[5]: 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 epochs = 100 batch_size = 100 input_shape = X_train.shape[1:4] receptive_field = 3 n_filters = 10 n_neurons_connected = 50 n_categories = 10 eta_vals = np.logspace(-5, 1, 7) lmbd_vals = np.logspace(-5, 1, 7) # ## Final part # In[6]: CNN_keras = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object) for i, eta in enumerate(eta_vals): for j, lmbd in enumerate(lmbd_vals): CNN = create_convolutional_neural_network_keras(input_shape, receptive_field, n_filters, n_neurons_connected, n_categories, eta, lmbd) CNN.fit(X_train, Y_train, epochs=epochs, batch_size=batch_size, verbose=0) scores = CNN.evaluate(X_test, Y_test) CNN_keras[i][j] = CNN print("Learning rate = ", eta) print("Lambda = ", lmbd) print("Test accuracy: %.3f" % scores[1]) print() # ## Final visualization # In[7]: # 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() # ## 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. # In[8]: 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 # ## 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. # In[10]: 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() # ## 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. # In[11]: 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. # ## 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. # In[13]: 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() # As you can see, our (4, 4, 64) outputs were flattened into vectors of shape (1024) before going through two Dense layers. # ## Compile and train the model # In[14]: 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)) # ## Finally, evaluate the model # In[15]: 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') test_loss, test_acc = model.evaluate(test_images, test_labels, verbose=2) print(test_acc) # ## Building our own CNN code # # Here we present a flexible and readable python code for a CNN # implemented with NumPy. We will present the code, showcase how to use # the codebase and fit a CNN that yields a 99% accuracy on the 28x28 # MNIST dataset within reasonable time. # # The CNN is compatible with all schedulers, cost functions and # activation functions discussed in constructing our neural network # codes. # # The CNN code consists of different types of Layer classes, including # Convolution2DLayer, Pooling2DLayer, FlattenLayer, FullyConnectedLayer # and OutputLayer, which can be added to the CNN object using the # interface of the CNN class. This allows you to easily construct your # own CNN, as well as allowing you to get used to an interface similar # to that of TensorFlow which is used for real world applications. # # Another important feature of this code is that it throws errors if # unreasonable decisions are made (for example using a kernel that is # larger than the image, not using a FlattenLayer, etc), and provides # the user with an informative error message. # ### List of contents: # # 1. Schedulers # # 2. Activation Functions # # 3. Cost Functions # # 4. Convolution # # 5. Layers # # 6. CNN # # 7. Some final remarks # ### Schedulers # # The code below shows object oriented implementations of the Constant, # Momentum, Adagrad, AdagradMomentum, RMS prop and Adam schedulers. All # of the classes belong to the shared abstract Scheduler class, and # share the update_change() and reset() methods allowing for any of the # schedulers to be seamlessly used during the training stage, as will # later be shown in the fit() method of the neural # network. Update_change() only has one parameter, the gradient # ($\delta^{l}_{j}a^{l-1}_k$), and returns the change which will be # subtracted from the weights. The reset() function takes no parameters, # and resets the desired variables. For Constant and Momentum, reset # does nothing. # In[14]: import autograd.numpy as np class Scheduler: """ Abstract class for Schedulers """ def __init__(self, eta): self.eta = eta # should be overwritten def update_change(self, gradient): raise NotImplementedError # overwritten if needed def reset(self): pass class Constant(Scheduler): def __init__(self, eta): super().__init__(eta) def update_change(self, gradient): return self.eta * gradient def reset(self): pass class Momentum(Scheduler): def __init__(self, eta: float, momentum: float): super().__init__(eta) self.momentum = momentum self.change = 0 def update_change(self, gradient): self.change = self.momentum * self.change + self.eta * gradient return self.change def reset(self): pass class Adagrad(Scheduler): def __init__(self, eta): super().__init__(eta) self.G_t = None def update_change(self, gradient): delta = 1e-8 # avoid division ny zero if self.G_t is None: self.G_t = np.zeros((gradient.shape[0], gradient.shape[0])) self.G_t += gradient @ gradient.T G_t_inverse = 1 / ( delta + np.sqrt(np.reshape(np.diagonal(self.G_t), (self.G_t.shape[0], 1))) ) return self.eta * gradient * G_t_inverse def reset(self): self.G_t = None class AdagradMomentum(Scheduler): def __init__(self, eta, momentum): super().__init__(eta) self.G_t = None self.momentum = momentum self.change = 0 def update_change(self, gradient): delta = 1e-8 # avoid division ny zero if self.G_t is None: self.G_t = np.zeros((gradient.shape[0], gradient.shape[0])) self.G_t += gradient @ gradient.T G_t_inverse = 1 / ( delta + np.sqrt(np.reshape(np.diagonal(self.G_t), (self.G_t.shape[0], 1))) ) self.change = self.change * self.momentum + self.eta * gradient * G_t_inverse return self.change def reset(self): self.G_t = None class RMS_prop(Scheduler): def __init__(self, eta, rho): super().__init__(eta) self.rho = rho self.second = 0.0 def update_change(self, gradient): delta = 1e-8 # avoid division ny zero self.second = self.rho * self.second + (1 - self.rho) * gradient * gradient return self.eta * gradient / (np.sqrt(self.second + delta)) def reset(self): self.second = 0.0 class Adam(Scheduler): def __init__(self, eta, rho, rho2): super().__init__(eta) self.rho = rho self.rho2 = rho2 self.moment = 0 self.second = 0 self.n_epochs = 1 def update_change(self, gradient): delta = 1e-8 # avoid division ny zero self.moment = self.rho * self.moment + (1 - self.rho) * gradient self.second = self.rho2 * self.second + (1 - self.rho2) * gradient * gradient moment_corrected = self.moment / (1 - self.rho**self.n_epochs) second_corrected = self.second / (1 - self.rho2**self.n_epochs) return self.eta * moment_corrected / (np.sqrt(second_corrected + delta)) def reset(self): self.n_epochs += 1 self.moment = 0 self.second = 0 # ### Usage of schedulers # # To initalize a scheduler, simply create the object and pass in the necessary parameters such as the learning rate and the momentum as shown below. As the Scheduler class is an abstract class it should not called directly, and will raise an error upon usage. # In[15]: momentum_scheduler = Momentum(eta=1e-3, momentum=0.9) adam_scheduler = Adam(eta=1e-3, rho=0.9, rho2=0.999) # Here is a small example for how a segment of code using schedulers could look. Switching out the schedulers is simple. # In[16]: weights = np.ones((3,3)) print(f"Before scheduler:\n{weights=}") epochs = 10 for e in range(epochs): gradient = np.random.rand(3, 3) change = adam_scheduler.update_change(gradient) weights = weights - change adam_scheduler.reset() print(f"\nAfter scheduler:\n{weights=}") # ### Cost functions # # In this section we will quickly look at cost functions that can be # used when creating the neural network. Every cost function takes the # target vector as its parameter, and returns a function valued only at # X such that it may easily be differentiated. # In[17]: def CostOLS(target): """ Return OLS function valued only at X, so that it may be easily differentiated """ def func(X): return (1.0 / target.shape[0]) * np.sum((target - X) ** 2) return func def CostLogReg(target): """ Return Logistic Regression cost function valued only at X, so that it may be easily differentiated """ def func(X): return -(1.0 / target.shape[0]) * np.sum( (target * np.log(X + 10e-10)) + ((1 - target) * np.log(1 - X + 10e-10)) ) return func def CostCrossEntropy(target): """ Return cross entropy cost function valued only at X, so that it may be easily differentiated """ def func(X): return -(1.0 / target.size) * np.sum(target * np.log(X + 10e-10)) return func # ### Usage of cost functions # # Below we will provide a short example of how these cost function may # be used to obtain results if you wish to test them out on your own # using AutoGrad's automatic differentiation. # In[18]: from autograd import grad target = np.array([[1, 2, 3]]).T a = np.array([[4, 5, 6]]).T cost_func = CostCrossEntropy cost_func_derivative = grad(cost_func(target)) valued_at_a = cost_func_derivative(a) print(f"Derivative of cost function {cost_func.__name__} valued at a:\n{valued_at_a}") # ### Activation functions # # Finally, before we look at the layers that make up the neural network, # we will look at the activation functions which can be specified # between the hidden layers and as the output function. Each function # can be valued for any given vector or matrix X, and can be # differentiated via derivate(). # In[19]: import autograd.numpy as np from autograd import elementwise_grad def identity(X): return X def sigmoid(X): try: return 1.0 / (1 + np.exp(-X)) except FloatingPointError: return np.where(X > np.zeros(X.shape), np.ones(X.shape), np.zeros(X.shape)) def softmax(X): X = X - np.max(X, axis=-1, keepdims=True) delta = 10e-10 return np.exp(X) / (np.sum(np.exp(X), axis=-1, keepdims=True) + delta) def RELU(X): return np.where(X > np.zeros(X.shape), X, np.zeros(X.shape)) def LRELU(X): delta = 10e-4 return np.where(X > np.zeros(X.shape), X, delta * X) def derivate(func): if func.__name__ == "RELU": def func(X): return np.where(X > 0, 1, 0) return func elif func.__name__ == "LRELU": def func(X): delta = 10e-4 return np.where(X > 0, 1, delta) return func else: return elementwise_grad(func) # ### Usage of activation functions # # Below we present a short demonstration of how to use an activation # function. The derivative of the activation function will be important # when calculating the output delta term during backpropagation. Note # that derivate() can also be used for cost functions for a more # generalized approach. # In[20]: z = np.array([[4, 5, 6]]).T print(f"Input to activation function:\n{z}") act_func = sigmoid a = act_func(z) print(f"\nOutput from {act_func.__name__} activation function:\n{a}") act_func_derivative = derivate(act_func) valued_at_z = act_func_derivative(a) print(f"\nDerivative of {act_func.__name__} activation function valued at z:\n{valued_at_z}") # ### Convolution # # In order to construct a convolutional neural network (CNN), it is # crucial to comprehend the fundamental principles of convolution and # how it aids in extracting information from images. Convolution, at its # core, is merely a mathematical operation between two functions that # yields another function. It is represented by an integral between two # functions, which is typically expressed as: # $$ # (f \ast g)(t):=\int_{-\infty}^{\infty} f(\tau) g(t-\tau) d \tau. # $$ # Here, f and g are the two functions on which we want to perform an # operation. The outcome of the convolution operation is represented by # $(f \ast g)$, and it is derived by sliding the function g over f and # computing the integral of their product at each position. If both # functions are continuous, convolution takes the form shown # above. However, if we discretize both f and g, the convolution # operation will take the form of a sum between the elements of f and g: # $$ # (f \ast g)[n]=\sum_{m=0}^{n-1} f[m] g[n-m]. # $$ # The key idea we utilize to extract the information contained in an # image is to slide an $m \times n$ matrix *g* over an $m \times n$ # matrix *f*. In our case, *f* represents the image, while *g* # represents the kernel, oftentimes called a filter. However, since our # convolution will be a two-dimensional variant, we need to extend our # mathematical formula with an additional summation: # $$ # (f \ast g)[i, j]\sum_{m=0}^{M-1}\sum_{n=0}^{N-1} f[m,n] g[i-m, j-n]. # $$ # It is imperative to note that the size of the kernel g is # significantly smaller than the size of the input image f, thereby # reducing the amount of computation necessary for feature # extraction. Furthermore, the kernel is usually a trainable parameter # in a convolutional neural network, allowing the network to learn # appropriate kernels for specific tasks. # # To give you an example of how 2D convolution works in practice, # suppose we have an image *f* of dimension $6 \times 6$ # $$ # f = \begin{bmatrix} # 4 & 1 & 2 & 9 & 8 & 6 \\ # 9 & 5 & 9 & 5 & 8 & 5 \\ # 1 & 5 & 9 & 7 & 6 & 4 \\ # 2 & 9 & 8 & 3 & 7 & 1 \\ # 8 & 1 & 6 & 4 & 2 & 2 \\ # 1 & 0 & 5 & 7 & 8 & 2 \\ # \end{bmatrix} # $$ # and a $3 \times 3$ kernel *g* called a low-pass filter. Note that the # kernel is usually rotated by 180 degrees during convolution, however # this has no effect on this kernel. # $$ # g = \frac{1}{9} # \begin{bmatrix} # 1 & 1 & 1 \\ # 1 & 1 & 1 \\ # 1 & 1 & 1 \\ # \end{bmatrix} # $$ # In order to filter the image, we have to extract a $3 \times 3$ # element from the upper left corner of *f*, and perform element-wise # multiplication of the extracted image pixels with the elements of the # kernel *g*: # $$ # \begin{bmatrix} # 4 & 1 & 2 \\ # 9 & 5 & 9 \\ # 1 & 5 & 9 \\ # \end{bmatrix} # \cdot # \begin{bmatrix} # \frac{1}{9} & \frac{1}{9} & \frac{1}{9} \\ # \frac{1}{9} & \frac{1}{9} & \frac{1}{9} \\ # \frac{1}{9} & \frac{1}{9} & \frac{1}{9} \\ # \end{bmatrix} # = # \begin{bmatrix} # \frac{4}{9} & \frac{1}{9} & \frac{2}{9} \\ # \frac{9}{9} & \frac{5}{9} & \frac{9}{9} \\ # \frac{1}{9} & \frac{5}{9} & \frac{9}{9} \\ # \end {bmatrix} # = \textbf{A} # $$ # Then, following the multiplication, we summarize all the elements of the resulting matrix A: # $$ # (f \ast g)[0, 0]= \sum_{i=0}^{2} \sum_{j=0}^{2} a_{i,j} = 5 # $$ # Which corresponds to the first element of the filtered image $(f \ast g)$. # # Here we use a stride of 1, a parameter denoted *s* which describes how # many indexes we move the kernel *g* to the right before repeating the # calculations above for the next $3 \times 3$ element of the image # *f*. It is usually presumed that *s*=1, however, larger values for *s* # can be used to reduce the dimentionality of the filtered image such # that the convolution operation is more computationally efficient. In # the context of a convolutional neural network, this will become very # useful. # # The full result of the convolution is: # $$ # (f \ast g) = # \begin{bmatrix} # 5 & 5.78 & 7 & 6.44 \\ # 6.33 & 6.67 & 6.89 & 5.11 \\ # 5.44 & 5.78 & 5.78 & 4 \\ # 4.44 & 4.78 & 5.56 & 4 \\ # \end{bmatrix} # $$ # The result is markedly smaller in shape than the original image. This occurs when using convolution without first padding the image with additional columns and rows, allowing us to keep the original image shape after sliding the kernel over the image. # How many rows and columns we wish to pad the image with depends strictly on the shape of the kernel, as we wish to pad the image with *r* additional rows and *c* additional columns. # $$ # r =\lfloor \frac{kernel\ height}{2} \rfloor \cdot 2 \\ # c =\lfloor \frac{kernel\ width}{2} \rfloor \cdot 2 # $$ # Note the notation $\lfloor \frac{kernel width}{2} \rfloor$ means that # we floor the result of the division, meaning we round down to a whole # number in case $\frac{kernel width}{2}$ results in a floating point # number. # # Using those simple equations, we find out by how much we have to # extend the dimensions of the original image. Before proceeding, # however, we might ask what we shall fill the additional rows and # columns with? One of the most common approaches to padding is # zero-padding, which as the name suggest, involves filling the rows and # columns with zeros. This is the approach that we will be using for # this demonstration. If we apply this padding to out original $6 \times 6$ # image, the result will be an $8 \times 8$ image as the kernel has a width and # height of 3. Note that the original image is encapsuled by the # zero-padded rows and columns: # $$ # \begin{bmatrix} # 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ # 0 & 4 & 1 & 2 & 9 & 8 & 6 & 0 \\ # 0 & 9 & 5 & 9 & 5 & 8 & 5 & 0 \\ # 0 & 1 & 5 & 9 & 7 & 6 & 4 & 0 \\ # 0 & 2 & 9 & 8 & 3 & 7 & 1 & 0 \\ # 0 & 8 & 1 & 6 & 4 & 2 & 2 & 0 \\ # 0 & 1 & 0 & 5 & 7 & 8 & 2 & 0 \\ # 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ # # \end{bmatrix} # $$ # Below we have provided code that demonstrates padding and convolution. As you will see when we run the code, the size of the image will remain unchanged when using padding.~ # In[21]: import numpy as np def padding(image, kernel): # calculate r and c r = (kernel.shape[0] // 2) * 2 c = (kernel.shape[1] // 2) * 2 # padded image dimensions padded_height = image.shape[0] + r padded_width = image.shape[1] + c # for more readable code k_half_height = kernel.shape[0] // 2 k_half_width = kernel.shape[1] // 2 # zero matrix with padded dimensions padded_img = np.zeros((padded_height, padded_width)) # place image into zero matrix padded_img[k_half_height : padded_height - k_half_height, k_half_width : padded_width - k_half_width] = image[:, :] return padded_img def convolve(original_image, padded_image, kernel, stride=1): # rotate kernel by 180 degrees kernel = np.rot90(np.rot90(kernel)) # note that kernel height // 2 is written as 'm' # and kernel width // 2 as 'n' in the mathematical notation m = kernel.shape[0] // 2 n = kernel.shape[1] // 2 r = (kernel.shape[0] // 2) * 2 c = (kernel.shape[1] // 2) * 2 # initialize output array convolved_image = np.zeros(original_image.shape) image_height = original_image.shape[0] image_width = original_image.shape[1] # the convolution for i in range(m, image_height + m, stride): for j in range(n, image_width + n, stride): convolved_image[i-m, j-n] = np.sum( padded_image[i : i + m, j : j + n] * kernel ) return convolved_image def convolve(image, kernel, stride=1): for i in range(2): kernel = np.rot90(kernel) k_half_height = kernel.shape[0] // 2 k_half_width = kernel.shape[0] // 2 conv_image = np.zeros(image.shape) pad_image = padding(image, kernel) for i in range(k_half_height, conv_image.shape[0] + k_half_height, stride): for j in range(k_half_width, conv_image.shape[1] + k_half_width, stride): conv_image[i - k_half_height, j - k_half_width] = np.sum( pad_image[ i - k_half_height : i + k_half_height + 1, j - k_half_width : j + k_half_width + 1 ] * kernel ) return conv_image # Fun fact: When filtering images, you will see that convolution involves rotating the kernel by 180 degrees. # However, this is not the case when applying convolution in a CNN, where the same operation not rotated by 180 degrees is called # cross-correlation. # In[22]: original_image = np.array([[4, 1, 2, 9, 8, 6], [9, 5, 9, 5, 8, 5], [1, 5, 9, 7, 6, 4], [2, 9, 8, 3, 7, 1], [8, 1, 6, 4, 2, 2], [1, 0, 5, 7, 8, 2]]) kernel = (1/9)*np.ones((3,3)) print(f"{original_image.shape=}") # note that convolve() performs padding convolved_image = convolve(original_image, kernel, stride=1) print(f"{convolved_image.shape=}") # As you can see, the resulting image is of the same size as the # original image. To round of our demonstration of convolution, we will # present the results of convolution using commonly used kernels. In a # CNN, the values of the kernels are randomly initialized, and then # learned during training. These kernels will extract information # regarding the picture, such as for example the edge detection filter # demonstrated below extracts the edges present in the picture. Of # course, there is no guarantee that the CNN will learn an edge # detection filter, but this should provide some intuiton as to how the # CNN is able to use kernels to make better predictions than a regular # feed forward neural network. # In[23]: # Now an example using a real image and first a gaussian low-pass filter and then a sobel filter import numpy as np import imageio.v3 as imageio import matplotlib.pyplot as plt import time def generate_gauss_mask(sigma, K=1): side = np.ceil(1 + 8 * sigma) y, x = np.mgrid[-side // 2 + 1 : (side // 2) + 1, -side // 2 + 1 : (side // 2) + 1] ker_coef = K / (2 * np.pi * sigma**2) g = np.exp(-((x**2 + y**2) / (2.0 * sigma**2))) return g, ker_coef img_path = "data/IMG-2167.JPG" image_of_cute_dog = imageio.imread(img_path, mode='L') plt.imshow(image_of_cute_dog, cmap="gray", vmin=0, vmax=255, aspect="auto") plt.title("Original image") plt.show() gauss, kernel = generate_gauss_mask(sigma=6) gauss_kernel = gauss*kernel filtered_image = convolve(image_of_cute_dog, gauss_kernel) plt.imshow(filtered_image, cmap="gray", vmin=0, vmax=255, aspect="auto") plt.title("Result of convolution with gauss kernel (blurring filter)") plt.show() sobel_kernel = np.array([[1, 2, 1], [0, 0, 0], [-1, -2, -1]]) filtered_image = convolve(image_of_cute_dog, sobel_kernel) plt.imshow(filtered_image, cmap="gray", vmin=0, vmax=255, aspect="auto") plt.title("Result of convolution with sobel kernel (edge detection filter)") plt.show() # ### Layers # # The code below initialises global variables for readability and # describes the abstract class Layers. This is not important in order to # understand the CNN, but is benefitial for organizing the code neatly. # In[24]: import math import autograd.numpy as np from copy import deepcopy, copy from autograd import grad from typing import Callable # global variables for index readability input_index = 0 node_index = 1 bias_index = 1 input_channel_index = 1 feature_maps_index = 1 height_index = 2 width_index = 3 kernel_feature_maps_index = 1 kernel_input_channels_index = 0 class Layer: def __init__(self, seed): self.seed = seed def _feedforward(self): raise NotImplementedError def _backpropagate(self): raise NotImplementedError def _reset_weights(self, previous_nodes): raise NotImplementedError # ### Convolution2DLayer: convolution in a hidden layer # # After establishing the foundational understanding of applying # convolution to spatial data, let us delve into the intricate workings # of a convolutional layer in a Convolutional Neural Network (CNN). The # primary function of convolution, as previously discussed, is to # extract pertinent information from images while simultaneously # decreasing the scale of our data. To initiate the image processing, we # shall begin by partitioning the images into color channels (unless the # image is grayscale), comprising three primary colors: red, green, and # blue. We will subsequently utilize trainable kernels to construct a # higher-dimensional encoding of each channel called feature # maps. Successive layers will receive these feature maps as inputs, # generating further encodings, albeit with reduced dimensions. The term # trainable kernels denotes the initialization of pre-defined # kernel-shaped weights, which we will then train via backpropagation, # similar to how weights are trained in a Feedforward Neural Network. # # To ensure seamless integration between our implementation of the # convolutional layer and popular machine learning frameworks like # Tensorflow (Keras) and PyTorch, we have adopted a design pattern that # mirrors the construction of models using these APIs. This involves # implementing our convolutional layer as a Python class or object, # which allows for a more modular and flexible approach to building # neural networks. By structuring our code in this way, users can easily # incorporate our implementation into their existing machine learning # pipelines without having to make significant changes to their # codebase. Additionally, this design pattern promotes code reusability # and makes it easier to maintain and update our convolutional layer # implementation over time. # # Note that the Convolution2DLayer takes in an activation function as a parameter, as it also performs non-linearity. # In[25]: class Convolution2DLayer(Layer): def __init__( self, input_channels, feature_maps, kernel_height, kernel_width, v_stride, h_stride, pad, act_func: Callable, seed=None, reset_weights_independently=True, ): super().__init__(seed) self.input_channels = input_channels self.feature_maps = feature_maps self.kernel_height = kernel_height self.kernel_width = kernel_width self.v_stride = v_stride self.h_stride = h_stride self.pad = pad self.act_func = act_func # such that the layer can be used on its own # outside of the CNN module if reset_weights_independently == True: self._reset_weights_independently() def _feedforward(self, X_batch): # note that the shape of X_batch = [inputs, input_maps, img_height, img_width] # pad the input batch X_batch_padded = self._padding(X_batch) # calculate height_index and width_index after stride strided_height = int(np.ceil(X_batch.shape[height_index] / self.v_stride)) strided_width = int(np.ceil(X_batch.shape[width_index] / self.h_stride)) # create output array output = np.ndarray( ( X_batch.shape[input_index], self.feature_maps, strided_height, strided_width, ) ) # save input and output for backpropagation self.X_batch_feedforward = X_batch self.output_shape = output.shape # checking for errors, no need to look here :) self._check_for_errors() # convolve input with kernel for img in range(X_batch.shape[input_index]): for chin in range(self.input_channels): for fmap in range(self.feature_maps): out_h = 0 for h in range(0, X_batch.shape[height_index], self.v_stride): out_w = 0 for w in range(0, X_batch.shape[width_index], self.h_stride): output[img, fmap, out_h, out_w] = np.sum( X_batch_padded[ img, chin, h : h + self.kernel_height, w : w + self.kernel_width, ] * self.kernel[chin, fmap, :, :] ) out_w += 1 out_h += 1 # Pay attention to the fact that we're not rotating the kernel by 180 degrees when filtering the image in # the convolutional layer, as convolution in terms of Machine Learning is a procedure known as cross-correlation # in image processing and signal processing # return a return self.act_func(output / (self.kernel_height)) def _backpropagate(self, delta_term_next): # intiate matrices delta_term = np.zeros((self.X_batch_feedforward.shape)) gradient_kernel = np.zeros((self.kernel.shape)) # pad input for convolution X_batch_padded = self._padding(self.X_batch_feedforward) # Since an activation function is used at the output of the convolution layer, its derivative # has to be accounted for in the backpropagation -> as if ReLU was a layer on its own. act_derivative = derivate(self.act_func) delta_term_next = act_derivative(delta_term_next) # fill in 0's for values removed by vertical stride in feedforward if self.v_stride > 1: v_ind = 1 for i in range(delta_term_next.shape[height_index]): for j in range(self.v_stride - 1): delta_term_next = np.insert( delta_term_next, v_ind, 0, axis=height_index ) v_ind += self.v_stride # fill in 0's for values removed by horizontal stride in feedforward if self.h_stride > 1: h_ind = 1 for i in range(delta_term_next.shape[width_index]): for k in range(self.h_stride - 1): delta_term_next = np.insert( delta_term_next, h_ind, 0, axis=width_index ) h_ind += self.h_stride # crops out 0-rows and 0-columns delta_term_next = delta_term_next[ :, :, : self.X_batch_feedforward.shape[height_index], : self.X_batch_feedforward.shape[width_index], ] # the gradient received from the next layer also needs to be padded delta_term_next = self._padding(delta_term_next) # calculate delta term by convolving next delta term with kernel for img in range(self.X_batch_feedforward.shape[input_index]): for chin in range(self.input_channels): for fmap in range(self.feature_maps): for h in range(self.X_batch_feedforward.shape[height_index]): for w in range(self.X_batch_feedforward.shape[width_index]): delta_term[img, chin, h, w] = np.sum( delta_term_next[ img, fmap, h : h + self.kernel_height, w : w + self.kernel_width, ] * np.rot90(np.rot90(self.kernel[chin, fmap, :, :])) ) # calculate gradient for kernel for weight update # also via convolution for chin in range(self.input_channels): for fmap in range(self.feature_maps): for k_x in range(self.kernel_height): for k_y in range(self.kernel_width): gradient_kernel[chin, fmap, k_x, k_y] = np.sum( X_batch_padded[ img, chin, h : h + self.kernel_height, w : w + self.kernel_width, ] * delta_term_next[ img, fmap, h : h + self.kernel_height, w : w + self.kernel_width, ] ) # all kernels are updated with weight gradient of kernel self.kernel -= gradient_kernel # return delta term return delta_term def _padding(self, X_batch, batch_type="image"): # same padding for images if self.pad == "same" and batch_type == "image": padded_height = X_batch.shape[height_index] + (self.kernel_height // 2) * 2 padded_width = X_batch.shape[width_index] + (self.kernel_width // 2) * 2 half_kernel_height = self.kernel_height // 2 half_kernel_width = self.kernel_width // 2 # initialize padded array X_batch_padded = np.ndarray( ( X_batch.shape[input_index], X_batch.shape[feature_maps_index], padded_height, padded_width, ) ) # zero pad all images in X_batch for img in range(X_batch.shape[input_index]): padded_img = np.zeros( (X_batch.shape[feature_maps_index], padded_height, padded_width) ) padded_img[ :, half_kernel_height : padded_height - half_kernel_height, half_kernel_width : padded_width - half_kernel_width, ] = X_batch[img, :, :, :] X_batch_padded[img, :, :, :] = padded_img[:, :, :] return X_batch_padded # same padding for gradients elif self.pad == "same" and batch_type == "grad": padded_height = X_batch.shape[height_index] + (self.kernel_height // 2) * 2 padded_width = X_batch.shape[width_index] + (self.kernel_width // 2) * 2 half_kernel_height = self.kernel_height // 2 half_kernel_width = self.kernel_width // 2 # initialize padded array delta_term_padded = np.zeros( ( X_batch.shape[input_index], X_batch.shape[feature_maps_index], padded_height, padded_width, ) ) # zero pad delta term delta_term_padded[ :, :, : X_batch.shape[height_index], : X_batch.shape[width_index] ] = X_batch[:, :, :, :] return delta_term_padded else: return X_batch def _reset_weights_independently(self): # sets seed to remove randomness inbetween runs if self.seed is not None: np.random.seed(self.seed) # initializes kernel matrix self.kernel = np.ndarray( ( self.input_channels, self.feature_maps, self.kernel_height, self.kernel_width, ) ) # randomly initializes weights for chin in range(self.kernel.shape[kernel_input_channels_index]): for fmap in range(self.kernel.shape[kernel_feature_maps_index]): self.kernel[chin, fmap, :, :] = np.random.rand( self.kernel_height, self.kernel_width ) def _reset_weights(self, previous_nodes): # sets weights self._reset_weights_independently() # returns shape of output used for subsequent layer's weight initiation strided_height = int( np.ceil(previous_nodes.shape[height_index] / self.v_stride) ) strided_width = int(np.ceil(previous_nodes.shape[width_index] / self.h_stride)) next_nodes = np.ones( ( previous_nodes.shape[input_index], self.feature_maps, strided_height, strided_width, ) ) return next_nodes / self.kernel_height def _check_for_errors(self): if self.X_batch_feedforward.shape[input_channel_index] != self.input_channels: raise AssertionError( f"ERROR: Number of input channels in data ({self.X_batch_feedforward.shape[input_channel_index]}) is not equal to input channels in Convolution2DLayerOPT ({self.input_channels})! Please change the number of input channels of the Convolution2DLayer such that they are equal" ) # ### Backpropagation in the convolutional layer # # As you may have noticed, we have not yet explained how the # backpropagation algorithm works in a convolutional layer. However, # having covered all other major details about convolutional layers, we # are now prepared to do so. It should come as no surprise that the # calculation of delta terms at each convolutional layer takes the form # of convolution. After the gradient has been propagated backwards # through the flattening layer, where it was reshaped into an # appropriate form, calculating the update value for the kernel is # simply a matter of convolving the output gradient with the input of # the layer for which we are updating the weights. For more detail, this # article serves as an excellent resource, see #