FYS-STK4155/doc/LectureNotes/_build/jupyter_execute/week44.ipynb

Week 44, Convolutional Neural Networks (CNN)

Morten Hjorth-Jensen, Department of Physics, University of Oslo, Norway

Date: October 28

Plan for week 44

Material for the lecture Monday October 28, 2024.

1. Convolutional Neural Networks

2. Readings and Videos:

• These lecture notes at https://github.com/CompPhysics/MachineLearning/blob/master/doc/pub/week44/ipynb/week44.ipynb

• For a more in depth discussion on neural networks we recommend Goodfellow et al chapter 9. See also chapter 11 and 12 on practicalities and applications

• Reading suggestions for implementation of CNNs see Rashcka et al.'s chapter 14 at https://github.com/rasbt/machine-learning-book/tree/main/ch14.

• Video on Deep Learning at https://www.youtube.com/playlist?list=PLZHQObOWTQDNU6R1_67000Dx_ZCJB-3pi

• Video on Convolutional Neural Networks from MIT at https://www.youtube.com/watch?v=iaSUYvmCekI&ab_channel=AlexanderAmini

• Video on CNNs from Stanford at https://www.youtube.com/watch?v=bNb2fEVKeEo&list=PLC1qU-LWwrF64f4QKQT-Vg5Wr4qEE1Zxk&index=6&ab_channel=StanfordUniversitySchoolofEngineering

• Video of lecture October 28 at https://youtu.be/rfrSfikAz94

• Whiteboard notes at https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/2024/NotesOctober28

Lab sessions on Tuesday and Wednesday

• Main focus is discussion of and work on project 2

• If you did not get time to finish the exercises from week 43, you can also keep working on them and hand in this coming Friday

Material for Lecture Monday October 28

Convolutional Neural Networks (recognizing images)

Convolutional neural networks (CNNs) were developed during the last decade of the previous century, with a focus on character recognition tasks. Nowadays, CNNs are a central element in the spectacular success of deep learning methods. The success in for example image classifications have made them a central tool for most machine learning practitioners.

CNNs are very similar to ordinary Neural Networks. They are made up of neurons that have learnable weights and biases. Each neuron receives some inputs, performs a dot product and optionally follows it with a non-linearity. The whole network still expresses a single differentiable score function: from the raw image 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

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.

Neural Networks vs CNNs

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.

Why CNNS for images, sound files, medical images from CT scans etc?

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:

• They are stored as multi-dimensional arrays (think of the pixels of a figure) .

• They feature one or more axes for which ordering matters (e.g., width and height axes for an image, time axis for a sound clip).

• One axis, called the channel axis, is used to access different views of the data (e.g., the red, green and blue channels of a color image, or the left and right channels of a stereo audio track).

These properties are not exploited when an affine transformation is applied; in fact, all the axes are treated in the same way and the topological information is not taken into account. Still, taking advantage of the implicit structure of the data may prove very handy in solving some tasks, like computer vision and speech recognition, and in these cases it would be best to preserve it. This is where discrete convolutions come into play.

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).

Regular NNs don’t scale well to full images

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.

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.

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).

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 remarks

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.

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)

A deep CNN model (From Raschka et al)

Figure 1: A deep CNN

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).

How to do image compression before the era of deep learning

The singular-value decomposition (SVD) algorithm has been for decades one of the standard ways of compressing images. The lectures on the SVD give many of the essential details concerning the SVD.

The orthogonal vectors which are obtained from the SVD, can be used to project down the dimensionality of a given image. In the example here we gray-scale an image and downsize it.

This recipe relies on us being able to actually perform the SVD. For large images, and in particular with many images to reconstruct, using the SVD may quickly become an overwhelming task. With the advent of efficient deep learning methods like CNNs and later generative methods, these methods have become in the last years the premier way of performing image analysis. In particular for classification problems with labelled images.

The SVD example

In [1]:

%matplotlib inline

from matplotlib.image import imread
import matplotlib.pyplot as plt
import scipy.linalg as ln
import numpy as np
import os
from PIL import Image
from math import log10, sqrt 
plt.rcParams['figure.figsize'] = [16, 8]
# Import image
A = imread(os.path.join("figslides/photo1.jpg"))
X = A.dot([0.299, 0.5870, 0.114]) # Convert RGB to grayscale
img = plt.imshow(X)
# convert to gray
img.set_cmap('gray')
plt.axis('off')
plt.show()
# Call image size
print(': %s'%str(X.shape))


# split the matrix into U, S, VT
U, S, VT = np.linalg.svd(X,full_matrices=False)
S = np.diag(S)
m = 800 # Image's width
n = 1200 # Image's height
j = 0
# Try compression with different k vectors (these represent projections):
for k in (5,10, 20, 100,200,400,500):
    # Original size of the image
    originalSize = m * n 
    # Size after compressed
    compressedSize = k * (1 + m + n) 
    # The projection of the original image
    Xapprox = U[:,:k] @ S[0:k,:k] @ VT[:k,:]
    plt.figure(j+1)
    j += 1
    img = plt.imshow(Xapprox)
    img.set_cmap('gray')
    
    plt.axis('off')
    plt.title('k = ' + str(k))
    plt.show() 
    print('Original size of image:')
    print(originalSize)
    print('Compression rate as Compressed image / Original size:')
    ratio = compressedSize * 1.0 / originalSize
    print(ratio)
    print('Compression rate is ' + str( round(ratio * 100 ,2)) + '%' )  
    # Estimate MQA
    x= X.astype("float")
    y=Xapprox.astype("float")
    err = np.sum((x - y) ** 2)
    err /= float(X.shape[0] * Xapprox.shape[1])
    print('The mean-square deviation '+ str(round( err)))
    max_pixel = 255.0
    # Estimate Signal Noise Ratio
    srv = 20 * (log10(max_pixel / sqrt(err)))
    print('Signa to noise ratio '+ str(round(srv)) +'dB')

: (2128, 3636)

Original size of image:
960000
Compression rate as Compressed image / Original size:
0.010421875
Compression rate is 1.04%
The mean-square deviation 993
Signa to noise ratio 18dB

Original size of image:
960000
Compression rate as Compressed image / Original size:
0.02084375
Compression rate is 2.08%
The mean-square deviation 590
Signa to noise ratio 20dB

Original size of image:
960000
Compression rate as Compressed image / Original size:
0.0416875
Compression rate is 4.17%
The mean-square deviation 341
Signa to noise ratio 23dB

Original size of image:
960000
Compression rate as Compressed image / Original size:
0.2084375
Compression rate is 20.84%
The mean-square deviation 70
Signa to noise ratio 30dB

Original size of image:
960000
Compression rate as Compressed image / Original size:
0.416875
Compression rate is 41.69%
The mean-square deviation 26
Signa to noise ratio 34dB

Original size of image:
960000
Compression rate as Compressed image / Original size:
0.83375
Compression rate is 83.38%
The mean-square deviation 6
Signa to noise ratio 41dB

Original size of image:
960000
Compression rate as Compressed image / Original size:
1.0421875
Compression rate is 104.22%
The mean-square deviation 3
Signa to noise ratio 43dB

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 operations (integration, summation etc) on two functions 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 and the process is commutative.

How can we use this? And what does it mean? Let us study some familiar examples first.

Convolution Examples: Polynomial multiplication

Our first example is that of a multiplication between two polynomials, which we will rewrite in terms of the mathematics of convolution. In the final stage, since the problem here is a discrete one, we will recast the final expression in terms of a matrix-vector multiplication, where the matrix is a so-called Toeplitz matrix .

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_0\beta_1\\
\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}.

Further simplification

Although we may have redundant operations with some few zeros for \beta_i, we can rewrite the above sum in a more compact way as

\delta_i = \sum_{k=0}^{k=m-1}\alpha_k\beta_{i-k},

where m=3 in our case, the maximum length of the vector \alpha. Note that the vector \boldsymbol{\beta} has length n=4. Below we will find an even more efficient representation.

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}.

Commutative process

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.

Toeplitz matrices

The above matrices are examples of so-called Toeplitz matrices. 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, 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.

Fourier series and Toeplitz matrices

This is an active and ogoing research area concerning CNNs. The following articles may be of interest

1. Read more about the convolution theorem and Fouriers series

2. Fourier Transform Layer

Generalizing the above one-dimensional case

In order to align the above simple case with the more general convolution cases, we rename \boldsymbol{\alpha}, whose length is m=3, with \boldsymbol{w}. We will interpret \boldsymbol{w} as a weight/filter function with which we want to perform the convolution with an input variable \boldsymbol{x} of length n. We will assume always that the filter \boldsymbol{w} has dimensionality m \le n.

We replace thus \boldsymbol{\beta} with \boldsymbol{x} and \boldsymbol{\delta} with \boldsymbol{y} and have

y(i)= \left(x*w\right)(i)= \sum_{k=0}^{k=m-1}w(k)x(i-k),

where m=3 in our case, the maximum length of the vector \boldsymbol{w}. Here the symbol * represents the mathematical operation of convolution.

Memory considerations

This expression leaves us however with some terms with negative indices, for example x(-1) and x(-2) which may not be defined. Our vector \boldsymbol{x} has components x(0), x(1), x(2) and x(3).

The index j for \boldsymbol{x} runs from j=0 to j=3 since \boldsymbol{x} is meant to represent a third-order polynomial.

Furthermore, the index i runs from i=0 to i=5 since \boldsymbol{y} contains the coefficients of a fifth-order polynomial. When i=5 we may also have values of x(4) and x(5) which are not defined.

Padding

The solution to this is what is called padding! We simply define a new vector x with two added elements set to zero before x(0) and two new elements after x(3) set to zero. That is, we augment the length of \boldsymbol{x} from n=4 to n+2P=8, where P=2 is the padding constant (a new hyperparameter), see discussions below as well.

New vector

We have a new vector defined as x(0)=0, x(1)=0, x(2)=\beta_0, x(3)=\beta_1, x(4)=\beta_2, x(5)=\beta_3, x(6)=0, and x(7)=0.

We have added four new elements, which are all zero. The benefit is that we can rewrite the equation for \boldsymbol{y}, with i=0,1,\dots,5,

y(i) = \sum_{k=0}^{k=m-1}w(k)x(i+(m-1)-k).

As an example, we have

y(4)=x(6)w(0)+x(5)w(1)+x(4)w(2)=0\times \alpha_0+\beta_3\alpha_1+\beta_2\alpha_2,

as before except that we have an additional term x(6)w(0), which is zero.

Similarly, for the fifth-order term we have

y(5)=x(7)w(0)+x(6)w(1)+x(5)w(2)=0\times \alpha_0+0\times\alpha_1+\beta_3\alpha_2.

The zeroth-order term is

y(0)=x(2)w(0)+x(1)w(1)+x(0)w(2)=\beta_0 \alpha_0+0\times\alpha_1+0\times\alpha_2=\alpha_0\beta_0.

Rewriting as dot products

If we now flip the filter/weight vector, with the following term as a typical example

y(0)=x(2)w(0)+x(1)w(1)+x(0)w(2)=x(2)\tilde{w}(2)+x(1)\tilde{w}(1)+x(0)\tilde{w}(0),

with \tilde{w}(0)=w(2), \tilde{w}(1)=w(1), and \tilde{w}(2)=w(0), we can then rewrite the above sum as a dot product of x(i:i+(m-1))\tilde{w} for element y(i), where x(i:i+(m-1)) is simply a patch of \boldsymbol{x} of size m-1.

The padding P we have introduced for the convolution stage is just another hyperparameter which is introduced as part of the architecture. Similarly, below we will also introduce another hyperparameter called Stride S.

Cross correlation

In essentially all applications one uses what is called cross correlation instead of the standard convolution described above. This means that multiplication is performed in the same direction and instead of the general expression we have discussed above (with infinite sums)

y(i) = \sum_{k=-\infty}^{k=\infty}w(k)x(i-k),

we have now

y(i) = \sum_{k=-\infty}^{k=\infty}w(k)x(i+k).

Both TensorFlow and PyTorch (as well as our own code example below), implement the last equation, although it is normally referred to as convolution. The same padding rules and stride rules discussed below apply to this expression as well.

We leave it as an exercise for you to convince yourself that the example we have discussed till now, gives the same final expression using the last expression.

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 X as input, we can have a filter defined by a two-dimensional kernel/weight/filter W. This leads to an output Y

Y(i,j)=(X * W)(i,j) = \sum_m\sum_n X(m,n)W(i-m,j-n).

Convolution is a commutative process, which means we can rewrite this equation as

Y(i,j)=(X * W)(i,j) = \sum_m\sum_n X(i-m,j-n)W(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.

As mentioned above, most deep learning libraries implement cross-correlation instead of convolution (although it is referred to as convolution)

Y(i,j)=(X * W)(i,j) = \sum_m\sum_n X(i+m,j+n)W(m,n).

CNNs in more detail, simple example

Let assume we have an input matrix X of dimensionality 3\times 3 and a 2\times 2 filter W given by the following matrices

\boldsymbol{X}=\begin{bmatrix}x_{00} & x_{01} & x_{02}  \\
                      x_{10} & x_{11} & x_{12}  \\
	              x_{20} & x_{21} & x_{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 X. We strongly recommend the repository on Arithmetic of deep learning by Dumoulin and Visin

Here we set the stride equal to S=1, which means that, starting with the element x_{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

Y_(i,j)=(X * W)(i,j) = \sum_m\sum_n X(i-m,j-n)W(m,n),

and obtain

\boldsymbol{Y}=\begin{bmatrix}x_{00}w_{00}+x_{01}w_{01}+x_{10}w_{10}+x_{11}w_{11} & x_{01}w_{00}+x_{02}w_{01}+x_{11}w_{10}+x_{12}w_{11}  \\
	              x_{10}w_{00}+x_{11}w_{01}+x_{20}w_{10}+x_{21}w_{11} & x_{11}w_{00}+x_{12}w_{01}+x_{21}w_{10}+x_{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{X}' of length 9 and a matrix \boldsymbol{W}' with dimension 4\times 9 as

\boldsymbol{X}'=\begin{bmatrix}x_{00} \\ x_{01} \\ x_{02} \\ x_{10} \\ x_{11} \\ x_{12} \\ x_{20} \\ x_{21} \\ x_{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{X}' is the same as the above convolution with stride S=1, that is

Y=(\boldsymbol{W}*\boldsymbol{X}),

is now given by \boldsymbol{W}'\boldsymbol{X}' which is a vector of length 4 instead of the originally resulting 2\times 2 output matrix.

The convolution stage

The convolution stage, where we apply different filters \boldsymbol{W} in order to reduce the dimensionality of an image, adds, in addition to the weights and biases (to be trained by the back propagation algorithm) that define the filters, two new hyperparameters, the so-called padding P and the stride S.

Finding the number of parameters

In the above example we have an input matrix of dimension $3\times 3$. In general we call the input for an input volume and it is defined by its width H_1, height H_1 and depth D_1. If we have the standard three color channels D_1=3.

The above example has W_1=H_1=3 and D_1=1.

When we introduce the filter we have the following additional hyperparameters

1. K the number of filters. It is common to perform the convolution of the input several times since by experience shrinking the input too fast does not work well

2. F as the filter's spatial extent

3. S as the stride parameter

4. P as the padding parameter

These parameters are defined by the architecture of the network and are not included in the training.

New image (or volume)

Acting with the filter on the input volume produces an output volume which is defined by its width W_2, its height H_2 and its depth D_2.

These are defined by the following relations

W_2 = \frac{(W_1-F+2P)}{S}+1,

Warning:

Output truncated. This notebook contains too many cells to display efficiently.