TD 1 : Image Classification

MOD 4.6 Deep Learning & Artificial Intelligence: an introduction

Introduction

The objective of this tutorial is to write a complete image classification program in Python. Two classification models will be successively developed and tested: k-nearest neighbors (KNN) and neural networks (NN).

Before you start

In this tutorial we use Python 3.7 or higher. Make sure it is properly installed. Make sure numpy is installed.

We assume that git is installed, and that you are familiar with the basic git commands. (Optionnaly, you can use GitHub Desktop.) We also assume that you have access to the ECL GitLab. If necessary, please consult this tutorial.

Important note: In this tutorial, you will be asked to write functions. If you want to test the correct operation of your functions in the same file, you must use the statement if __name__ == "__main__"::

def func():
    ...

if __name__ == "__main__":
    func()

README

Your repository must contain a text file README.md that introduces the project in a short way. To learn how to write a readme, visit https://www.makeareadme.com. We recommend that you include at least the sections Description and Usage.

Prepare your directory

1. Create a new blank project on the ECL GitLab (New project then Create blank project).
2. Fill in the form as follows.
   • Project name: Image classification.
   • Project slug: image-classification.
   • Visibility Level: private (don't forget to add your teacher as member (developper) so that he can have access to the project)
   • Project Configuration: Initialize repository with a README
3. Clone the repository.

   git clone https://gitlab.ec-lyon.fr/<user>/image-classification

4. In this tutorial you will use files that should not be pushed to the remote repository. To ignore them when committing, you can put their path in a file named .gitignore. For simplicity, we use the .gitignore file recommended by GitHub for Python projects.

Prepare the CIFAR dataset

The image database used for the experiments is CIFAR-10 which consists of 60 000 color images of size 32x32 divided into 10 classes (plane, car, bird, cat, ...). This database can be obtained at the address https://www.cs.toronto.edu/~kriz/cifar.html where are also given the indications to read the data.

1. Create a folder named data in which you move the downloaded cifar-10-batches-py folder. Make sure that the data folder is ignored when commiting.
2. Create a Python file named read_cifar.py. Write the function read_cifar_batch taking as parameter the path of a single batch as a string, and returning:
   • a matrix data of size (batch_size x data_size) where batch_size is the number of available data in the batch, and data_size the dimension of these data (number of numerical values describing the data), and
   • a vector labels of size batch_size whose values correspond to the class code of the data of the same index in data. data must be np.float32 array and labels must be np.int64 array.
3. Write the function read_cifar taking as parameter the path of the directory containing the six batches (five data_batch and one test_batch) as a string, and returning
   • a matrix data of shape (batch_size x data_size) where batch_size is the number of available data in all batches (including test_batch), and
   • a vector labels of size batch_size whose values correspond to the class code of the data of the same index in data. data must be np.float32 array and labels must be np.int64 array.
4. Write the function split_dataset which splits the dataset into a training set and a test set. The data must be shuffled, so that two successive calls shouldn't give the same output. This function takes as parameter
   • data and labels, two arrays that have the same size in the first dimension.
   • split, a float between 0 and 1 which determines the split factor of the training set with respect to the test set. This function must return
   • data_train the training data,
   • labels_train the corresponding labels,
   • data_test the testing data, and
   • labels_test the corresponding labels.

k-nearest neighbors

1. Create a Python fil named knn.py. Write the function distance_matrix taking as parameters two matrices and returns dists, the L2 Euclidean distance matrix. The computation must be done only with matrix manipulation (no loops). Hint:

   (a-b)^2 = a^2 + b^2 - 2 ab

2. Write the function knn_predict taking as parameters:

   • dists the distance matrix between the train set and the test set,
   • labels_train the training labels, and
   • k the number of of neighbors. This function must return the predicted labels for the elements of data_test.

   Note: if the memory occupation is too important, you can use several batches for the calculation of the distance matrix (loop on sub-batches of test data).

3. Write the function evaluate_knn taking as parameters:

   • data_train the training data,
   • labels_train the corresponding labels,
   • data_test the testing data,
   • labels_test the corresponding labels, and
   • k the number of of neighbors. This function must return the classification rate (accuracy).

4. For split_factor=0.9, plot the variation of the accuracy as a function of k (from 1 to 20). Save the plot as an image named knn.png in the directory results.

Artificial Neural Network

The objective here is to develop a classifier based on a multilayer perceptron (MLP) neural network.

First of all, let's focus on the backpropagation of the gradient with an example. If you still have trouble understanding the intuition behind the back propagation of the gradient, check out this video: 3Blue1Brown/Backpropagation calculus | Chapter 4, Deep learning.

The weight matrix of the layer

1. We admit that for the sigmoide function
   \sigma
   , the derivative is
   \sigma' = \sigma \times (1-\sigma)
2. Express
   \frac{\partial C}{\partial A^{(2)}}
   , i.e. the vector of
   \frac{\partial C}{\partial a^{(2)}_i}
   as a function of
   A^{(2)}
   and
   Y
   .
3. Using the chaining rule, express
   \frac{\partial C}{\partial Z^{(2)}}
   , i.e. the vector of
   \frac{\partial C}{\partial z^{(2)}_i}
   as a function of
   \frac{\partial C}{\partial A^{(2)}}
   and
   A^{(2)}
   .
4. Similarly, express
   \frac{\partial C}{\partial W^{(2)}}
   , i.e. the matrix of
   \frac{\partial C}{\partial w^{(2)}_{i,j}}
   as a function of
   \frac{\partial C}{\partial Z^{(2)}}
   and
   A^{(1)}
   .
5. Similarly, express
   \frac{\partial C}{\partial B^{(2)}}
   as a function of
   \frac{\partial C}{\partial Z^{(2)}}
   .
6. Similarly, express
   \frac{\partial C}{\partial A^{(1)}}
   as a function of
   \frac{\partial C}{\partial Z^{(2)}}
   and
   W^{(2)}
   .
7. Similarly, express
   \frac{\partial C}{\partial Z^{(1)}}
   as a function of
   \frac{\partial C}{\partial A^{(1)}}
   and
   A^{(1)}
   .
8. Similarly, express
   \frac{\partial C}{\partial W^{(1)}}
   as a function of
   \frac{\partial C}{\partial Z^{(1)}}
   and
   A^{(0)}
   .
9. Similarly, express
   \frac{\partial C}{\partial B^{(1)}}
   as a function of
   \frac{\partial C}{\partial Z^{(1)}}
   .

Below is a Python code performing a forward pass and computing the cost in a network containing one hidden layer for the regression task. It uses the sigmoid function as the activation function:

import numpy as np

N = 30  # number of input data
d_in = 3  # input dimension
d_h = 3  # number of neurons in the hidden layer
d_out = 2  # output dimension (number of neurons of the output layer)

# Random initialization of the network weights and biaises
w1 = 2 * np.random.rand(d_in, d_h) - 1  # first layer weights
b1 = np.zeros((1, d_h))  # first layer biaises
w2 = 2 * np.random.rand(d_h, d_out) - 1  # second layer weights
b2 = np.zeros((1, d_out))  # second layer biaises

data = np.random.rand(N, d_in)  # create a random data
targets = np.random.rand(N, d_out)  # create a random targets

# Forward pass
a0 = data # the data are the input of the first layer
z1 = np.matmul(a0, w1) + b1  # input of the hidden layer
a1 = 1 / (1 + np.exp(-z1))  # output of the hidden layer (sigmoid activation function)
z2 = np.matmul(a1, w2) + b2  # input of the output layer
a2 = 1 / (1 + np.exp(-z2))  # output of the output layer (sigmoid activation function)
predictions = a2  # the predicted values are the outputs of the output layer

# Compute loss (MSE)
loss = np.mean(np.square(predictions - targets))
print(loss)

10. Create a Python file named mlp.py. Use the above code to write the function learn_once_mse taking as parameters:

    • w1, b1, w2 and b2 the weights and biases of the network,
    • data a matrix of shape (batch_size x d_in),
    • targets a matrix of shape (batch_size x d_out),
    • learning_rate the learning rate,

    that perform one gradient descent step, and returns:

    • w1, b1, w2 and b2 the updated weights and biases of the network,
    • loss the loss, for monitoring purpose.

MSE loss is not well suited for a classification task. Instead, we want to use the binary cross-entropy loss. To use this loss, we need the target to be is the one-hot encoding of the desired labels. Example:

one_hot(labels=[1 2 0]) = [[0 1 0]
                           [0 0 1]
                           [1 0 0]]

We also need that the last activation layer of the network to be a softmax layer.

11. Write the function one_hot taking a (n)-D array as parameters and returning the corresponding (n+1)-D one-hot matrix.

12. Write the function learn_once_cross_entropy taking as parameters:

    • w1, b1, w2 and b2 the weights and biases of the network,
    • data a matrix of shape (batch_size x d_in),
    • labels_train a vector of size batch_size, and
    • learning_rate the learning rate,

    that perform one gradient descent step using a binary cross-entropy loss. We admit that

    \frac{\partial C}{\partial Z^{(2)}} = A^{(2)} - Y
    , where
    Y
    is a one-hot vector encoding the label. The function must return:
    • w1, b1, w2 and b2 the updated weights and biases of the network,
    • loss the loss, for monitoring purpose.

13. Write the function train_mlp taking as parameters:

    • w1, b1, w2 and b2 the weights and biases of the network,
    • data_train a matrix of shape (batch_size x d_in),
    • labels_train a vector of size batch_size,
    • learning_rate the learning rate, and
    • num_epoch the number of training epoch,

    that perform num_epoch of training steps and returns:

    • w1, b1, w2 and b2 the updated weights and biases of the network,
    • train_accuracies the list of train accuracies across epochs as a list of floats.

14. Write the function test_mlp taking as parameters:

    • w1, b1, w2 and b2 the weights and biases of the network,
    • data_test a matrix of shape (batch_size x d_in), and
    • labels_test a vector of size batch_size,

    testing the network on the test set and returns:

    • test_accuracy the testing accuracy.

15. Write the function run_mlp_training taking as parameter:

    • data_train, labels_train, data_test, labels_test, the training and testing data,
    • d_h the number of neurons in the hidden layer
    • learning_rate the learning rate, and
    • num_epoch the number of training epoch,

    that train an MLP classifier and return the training accuracies across epochs as a list of floats and the final testing accuracy as a float.

16. For split_factor=0.9, d_h=64, learning_rate=0.1 and num_epoch=100, plot the evolution of learning accuracy across learning epochs. Save the graph as an image named mlp.png in the results directory.

To go further (optional)

Unittest

Your code should contain unit tests. All unit tests should be contained in the tests directory located at the root of the directory. We choose to use pytest. To help you write unit tests, you can consult the pytest documentation.

Deep dive into the classifier

Experiments will have to be carried out by studying the following variations:

• use image representation by descriptors (LBP, HOG, ...) instead of raw pixels using the scikit-image module.
• use of N-fold cross-validation instead of a fixed learning and testing subset.

To be handed in

This work must be done individually. The expected output is a repository on https://gitlab.ec-lyon.fr. It must contain the complete, minimal and functional code corresponding to this tutorial.

The last commit is due before 11:59 pm on Wednesday, November 20, 2024. Subsequent commits will not be considered.