MNIST Handwritten Digit Classifier

An implementation of multilayer neural network using numpy library. The implementation is a modified version of Michael Nielsen's implementation in Neural Networks and Deep Learning book.

Brief Background:

If you are familiar with basics of Neural Networks, feel free to skip this section. For total beginners who landed up here before reading anything about Neural Networks:

Sigmoid Neuron

Neural networks are made up of building blocks known as Sigmoid Neurons. These are named so because their output follows Sigmoid Function.
x_j are inputs, which are weighted by w_j weights and the neuron has its intrinsic bias b. The output of neuron is known as "activation ( a )".

Note: There are other functions in use other than sigmoid, but this information for now is sufficient for beginners.

A neural network is made up by stacking layers of neurons, and is defined by the weights of connections and biases of neurons. Activations are a result dependent on a certain input.

Why a modified implementation ?

This book and Stanford's Machine Learning Course by Prof. Andrew Ng are recommended as good resources for beginners. At times, it got confusing to me while referring both resources:

MATLAB has 1-indexed data structures, while numpy has them 0-indexed. Some parameters of a neural network are not defined for the input layer, so there was a little mess up in mathematical equations of book, and indices in code. For example according to the book, the bias vector of second layer of neural network was referred as bias[0] as input layer (first layer) has no bias vector. I found it a bit inconvenient to play with.

I am fond of Scikit Learn's API style, hence my class has a similar structure of code. While theoretically it resembles the book and Stanford's course, you can find simple methods such as fit, predict, validate to train, test, validate the model respectively.

Naming and Indexing Convention:

I have followed a particular convention in indexing quantities. Dimensions of quantities are listed according to this figure.

Small Labelled Neural Network

Layers

Input layer is the 0^th layer, and output layer is the L^th layer. Number of layers: N_L = L + 1.
```
sizes = [2, 3, 1]
```

Weights

Weights in this neural network implementation are a list of matrices (numpy.ndarrays). weights[l] is a matrix of weights entering the l^th layer of the network (Denoted as w^l).
An element of this matrix is denoted as w^l_jk. It is a part of j^th row, which is a collection of all weights entering j^th neuron, from all neurons (0 to k) of (l-1)^th layer.
No weights enter the input layer, hence weights[0] is redundant, and further it follows as weights[1] being the collection of weights entering layer 1 and so on.
```
weights = |¯   [[]],    [[a, b],    [[p],   ¯|
      |              [c, d],     [q],    |
      |_             [e, f]],    [r]]   _|
```

Biases

Biases in this neural network implementation are a list of one-dimensional vectors (numpy.ndarrays). biases[l] is a vector of biases of neurons in the l^th layer of network (Denoted as b^l).
An element of this vector is denoted as b^l_j. It is a part of j^th row, the bias of j^th in layer.
Input layer has no biases, hence biases[0] is redundant, and further it follows as biases[1] being the biases of neurons of layer 1 and so on.
```
biases = |¯   [[],    [[0],    [[0]]   ¯|
     |     []],    [1],             |
     |_            [2]],           _|
```

'Z's

For input vector x to a layer l, z is defined as: z^l = w^l . x + b^l
Input layer provides x vector as input to layer 1, and itself has no input, weight or bias, hence zs[0] is redundant.
Dimensions of zs will be same as biases.

Activations

Activations of l^th layer are outputs from neurons of l^th which serve as input to (l+1)^th layer. The dimensions of biases, zs and activations are similar.
Input layer provides x vector as input to layer 1, hence activations[0] can be related to x - the input training example.

Execution of Neural network

#to train and test the neural network algorithm, please use the following command
python main.py