Machine learning note 8: octave for handwritten digit recognition based on Neural Network

Problems to be solved （3 individual ）:

1. Data loading and visualization

%  Instructions
%  ------------
%
%  This file contains code that helps you get started on the
%  linear exercise. You will need to complete the following functions
%  in this exericse:
%
%     sigmoidGradient.m
%     randInitializeWeights.m
%     nnCostFunction.m
%
%  You will not need to change any code in this file,
%  or any other files other than those mentioned above.
%


%% Initialization
clear ;
close all;
clc

%% Setup the parameters you will use for this exercise
input_layer_size  = 400;  % 20x20 Input Images of Digits
hidden_layer_size = 25;   % 25 hidden units
num_labels = 10;          % 10 labels, from 1 to 10
                          % (note that we have mapped "0" to label 10)

%% =========== Part 1: Loading and Visualizing Data =============
%  We start the exercise by first loading and visualizing the dataset.
%  You will be working with a dataset that contains handwritten digits.
%

% Load Training Data
fprintf('Loading and Visualizing Data ...\n')

% X has 5000 samples and 400 characteristics.
load('ex4data1.mat');
m = size(X, 1);

% Randomly select 100 data points to display.
sel = randperm(size(X, 1));
sel = sel(1:100);

% The starting point is (1, 1), not (0, 0).
% K is a matrix of 100*400
k = X(sel,:);
displayData(k);

fprintf('Program paused. Press enter to continue.\n');

1. sel Array （1*100）：

2. Digital Visualization ：

1.1 displayData.m

Realization function ： Display a two-dimensional array in the grid , And automatically generate a width for the number in each grid （ Width if any , It's not generated ）

function [h, display_array] = displayData(X, example_width)
%DISPLAYDATA Display 2D data in a nice grid
%   [h, display_array] = DISPLAYDATA(X, example_width) displays 2D data
%   stored in X in a nice grid. It returns the figure handle h and the
%   displayed array if requested.


%% Set the length and width of the image, equal by default that is 20 and 20
% Set example_width automatically if not passed in
if ~exist('example_width', 'var') || isempty(example_width)
	example_width = round(sqrt(size(X, 2))); % 2 is columns;
                                           % example_width is 20
end

% Gray Image
colormap(gray);

% Compute rows, cols;
% example_height is 20
[m n] = size(X);
example_height = (n / example_width);


%% Compute number of items to display
% .._rows*.._cols is 10 * 1
display_rows = floor(sqrt(m));
display_cols = ceil(m / display_rows);

%% Between images padding
% Separated by a black line
pad = 1; % What's the meaning of this?

% Setup blank display;
% display_array is (1+10*21) * (1+1*21)
display_array = - ones(pad + display_rows * (example_height + pad), ...
                       pad + display_cols * (example_width + pad));

% Copy each example into a patch on the display_array
curr_ex = 1;
for j = 1:display_rows
	for i = 1:display_cols
		if curr_ex > m,
			break;
		end
		% Copy the patch

		% Get the max value of the patch and normalize each sample
		max_val = max(abs(X(curr_ex, :)));
		display_array(pad + (j - 1) * (example_height + pad) + (1:example_height), ...
		              pad + (i - 1) * (example_width + pad) + (1:example_width)) = ...
            % reshape a (1*400) row vector into a (example_height*example_width)
            % square matrix.
						reshape(X(curr_ex, :), example_height, example_width) / max_val;
		curr_ex = curr_ex + 1;
	end
	if curr_ex > m,
		break;
	end
end

% Display Image
h = imagesc(display_array, [-1 1]);

% Do not show axis
axis image off

drawnow;

end

Code flow chart ：

2. Parameter loading

% Load some pre-initialized neural network parameters.

fprintf('\nLoading Saved Neural Network Parameters ...\n')

% Load the weights into variables Theta1 and Theta2
load('ex4weights.mat');

% Unroll parameters
% Theta1:25*401=10025
% Theta2:10*26=260
% Expand by column,10285*1
nn_params = [Theta1(:) ; Theta2(:)];

3. Calculate the cost based on forward propagation algorithm

%  To the neural network, you should first start by implementing the
%  feedforward part of the neural network that returns the cost only. You
%  should complete the code in nnCostFunction.m to return cost. After
%  implementing the feedforward to compute the cost, you can verify that
%  your implementation is correct by verifying that you get the same cost
%  as us for the fixed debugging parameters.
%
%  We suggest implementing the feedforward cost *without* regularization
%  first so that it will be easier for you to debug. Later, in part 4, you
%  will get to implement the regularized cost.
%
fprintf('\nFeedforward Using Neural Network ...\n')

% Weight regularization parameter (we set this to 0 here).
% *without* regularization first
lambda = 0;

J = nnCostFunction(nn_params, input_layer_size, hidden_layer_size, ...
                   num_labels, X, y, lambda);

fprintf(['Cost at parameters (loaded from ex4weights): %f '...
         '\n(this value should be about 0.287629)\n'], J);

fprintf('\nProgram paused. Press enter to continue.\n');

The result of non regularization ：

4. Regularization

%  Once your cost function implementation is correct, you should now
%  continue to implement the regularization with the cost.
%

fprintf('\nChecking Cost Function (w/ Regularization) ... \n')

% Weight regularization parameter (we set this to 1 here).
lambda = 1;

J_reg = nnCostFunction(nn_params, input_layer_size, hidden_layer_size, ...
                   num_labels, X, y, lambda);

fprintf(['Cost at parameters (loaded from ex4weights): %f '...
         '\n(this value should be about 0.383770)\n'], J);

fprintf('Program paused. Press enter to continue.\n');

Regularization results ：

4.1 nnCostFunction.m

function [J grad] = nnCostFunction(nn_params, ...
                                   input_layer_size, ...
                                   hidden_layer_size, ...
                                   num_labels, ...
                                   X, y, lambda)
%NNCOSTFUNCTION Implements the neural network cost function for a two layer
%neural network which performs classification
%   [J grad] = NNCOSTFUNCTON(nn_params, hidden_layer_size, num_labels, ...
%   X, y, lambda) computes the cost and gradient of the neural network. The
%   parameters for the neural network are "unrolled" into the vector
%   nn_params and need to be converted back into the weight matrices.
%
%   The returned parameter grad should be a "unrolled" vector of the
%   partial derivatives of the neural network.
%

% Reshape nn_params back into the parameters Theta1 and Theta2, the weight matrices
% for our 2 layer neural network
%  Column vector to be expanded , Remodel as theta matrix 
% Theta Reshaped matrix ： First dimension  =  The number of features of the next layer , The second dimension  =  Number of features of the first layer +1
Theta1 = reshape(nn_params(1:hidden_layer_size * (input_layer_size + 1)), ...
                 hidden_layer_size, (input_layer_size + 1));

Theta2 = reshape(nn_params((1 + (hidden_layer_size * (input_layer_size + 1))):end), ...
                 num_labels, (hidden_layer_size + 1));

% Setup some useful variables
%  Set the number of samples , Here for 5000 individual 
m = size(X, 1);

% You need to return the following variables correctly
J = 0;
Theta1_grad = zeros(size(Theta1));
Theta2_grad = zeros(size(Theta2));

part1：
principle ：

% Instructions: You should complete the code by working through the
%               following parts.

% Part 1: Feedforward the neural network and return the cost in the
%         variable J. After implementing Part 1, you can verify that your
%         cost function computation is correct by verifying the cost
%         computed in ex4.m
%

%  The first layer calculates （ Input layer to hidden layer ）, Add bias number （+1）, Calculation sigmoid The value in 
X = [ones(m,1) X];      % 5000 * 401
net_h = X * Theta1'; % 5000 * 25 out_h = sigmoid(net_h); %  The second layer calculates （ Hidden layer to output layer ） out_h = [ones(m,1) out_h]; % 5000*26 net_o = out_h * Theta2';   % 5000*10
out_o = sigmoid(net_o);

% out_o by h（x） function , Dimension for 5000*10,y The dimensions are 5000*1, To put y The dimension of is converted to 5000*10, I.e out_o The dimensions are the same 
for i =1:num_labels,
  matrices_y(:,i) = (y==i);
end
matrices_J = log(out_o) .* matrices_y +log((1 - out_o)) .* (1 - matrices_y);
J = (-1/m)*sum(sum(matrices_J));

part2： Back propagation algorithm implementation （ Regular terms are not considered ）

% Part 2: Implement the backpropagation algorithm to compute the gradients
%         Theta1_grad and Theta2_grad. You should return the partial derivatives of
%         the cost function with respect to Theta1 and Theta2 in Theta1_grad and
%         Theta2_grad, respectively. After implementing Part 2, you can check
%         that your implementation is correct by running checkNNGradients
%
%         Note: The vector y passed into the function is a vector of labels
%               containing values from 1..K. You need to map this vector into a
%               binary vector of 1's and 0's to be used with the neural network
%               cost function.
%
%         Hint: We recommend implementing backpropagation using a for-loop
%               over the training examples if you are implementing it for the
%               first time.
%

error_termOut = zeros(m, num_labels);

%  Error of the third output layer 
error_termOut = out_o-matrices_y   % 5000*10

%  Error of the second hidden layer 
% error_termH The dimension of is consistent with the dimension of the second layer 
error_termH = (error_termOut*Theta2).*out_h.*(1-out_h); % 5000*26（ Add offset term +1）

Theta2_grad = error_termOut'*out_h % error_termOut Dimension for 5000*10 % out_h Dimension for 5000*26 %  Forward propagation algorithm adds bias term , Backward propagation , Get rid of the first column  Theta1_grad = error_termH(:,2:end)'*X %Theta1_grad Dimension for 25*401

disp(Theta1_grad);

disp(Theta2_grad);

part3： Consider the regular term
First , Calculation Theta1_reg、Theta2_grad：
Be careful ： Calculate the regular term from the second column

Last , Calculate regular term ：

% Part 3: Implement regularization with the cost function and gradients.
%
%         Hint: You can implement this around the code for
%               backpropagation. That is, you can compute the gradients for
%               the regularization separately and then add them to Theta1_grad
%               and Theta2_grad from Part 2.
%

Theta2_reg = Theta2_grad(:,2:end)+lambda.*Theta2(:,2:end);
Theta2_grad = (1/m).*[Theta2_grad(:,1) Theta2_reg];

Theta1_reg = Theta1_grad(:,2:end)+lambda.*Theta1(:,2:end);
Theta1_grad = (1/m).*[Theta1_grad(:,1) Theta1_reg];

%  Plus the regular term 
reg1 = sum(sum(Theta1(:,2:end).^2));
reg2 = sum(sum(Theta2(:,2:end).^2));

reg = (lambda/2*m)*(reg1+reg2);    %  Be careful 2m It doesn't mean 2*m
J = J +reg;

% -------------------------------------------------------------

% =========================================================================

% Unroll gradients
grad = [Theta1_grad(:) ; Theta2_grad(:)];

end

5. Gradient descent function

%% ================ Part 5: Sigmoid Gradient  ================
%  Before you start implementing the neural network, you will first
%  implement the gradient for the sigmoid function. You should complete the
%  code in the sigmoidGradient.m file.
%

fprintf('\nEvaluating sigmoid gradient...\n')

g = sigmoidGradient([-1 -0.5 0 0.5 1]);
fprintf('Sigmoid gradient evaluated at [-1 -0.5 0 0.5 1]:\n ');
fprintf('%f ', g);
fprintf('\n\n');

fprintf('Program paused. Press enter to continue.\n');

5.1 sigmoid.m

function g = sigmoid(z)
%SIGMOID Compute sigmoid functoon
%   J = SIGMOID(z) computes the sigmoid of z.

g = 1.0 ./ (1.0 + exp(-z));
end

5.2 sigmoidGradient.m

function g = sigmoidGradient(z)
%SIGMOIDGRADIENT returns the gradient of the sigmoid function
%evaluated at z
%   g = SIGMOIDGRADIENT(z) computes the gradient of the sigmoid function
%   evaluated at z. This should work regardless if z is a matrix or a
%   vector. In particular, if z is a vector or matrix, you should return
%   the gradient for each element.

g = zeros(size(z));

% ====================== YOUR CODE HERE ======================
% Instructions: Compute the gradient of the sigmoid function evaluated at
%               each value of z (z can be a matrix, vector or scalar).

g = sigmoid(z).*(1-sigmoid(z))

% =============================================================

end

6. Initialize parameters

%% ================ Part 6: Initializing Pameters ================
%  In this part of the exercise, you will be starting to implment a two
%  layer neural network that classifies digits. You will start by
%  implementing a function to initialize the weights of the neural network
%  (randInitializeWeights.m)

fprintf('\nInitializing Neural Network Parameters ...\n')

initial_Theta1 = randInitializeWeights(input_layer_size, hidden_layer_size);
initial_Theta2 = randInitializeWeights(hidden_layer_size, num_labels);

% Unroll parameters
initial_nn_params = [initial_Theta1(:) ; initial_Theta2(:)];

6.1 randInitializeWeights.m

function W = randInitializeWeights(L_in, L_out)
%RANDINITIALIZEWEIGHTS Randomly initialize the weights of a layer with L_in
%incoming connections and L_out outgoing connections
%   W = RANDINITIALIZEWEIGHTS(L_in, L_out) randomly initializes the weights
%   of a layer with L_in incoming connections and L_out outgoing
%   connections.
%
%   Note that W should be set to a matrix of size(L_out, 1 + L_in) as
%   the first column of W handles the "bias" terms
%

% You need to return the following variables correctly
W = zeros(L_out, 1 + L_in);

% ====================== YOUR CODE HERE ======================
% Instructions: Initialize W randomly so that we break the symmetry while
%               training the neural network.
%
% Note: The first column of W corresponds to the parameters for the bias unit
%

epsilon = 0.01;
w = (2*epsilon) .* rand(L_out,L_in) - epsilon;


% =========================================================================

end

7. Back propagation algorithm implementation

%% =============== Part 7: Implement Backpropagation ===============
%  Once your cost matches up with ours, you should proceed to implement the
%  backpropagation algorithm for the neural network. You should add to the
%  code you've written in nnCostFunction.m to return the partial % derivatives of the parameters. % fprintf('\nChecking Backpropagation... \n'); % Check gradients by running checkNNGradients checkNNGradients; fprintf('\nProgram paused. Press enter to continue.\n');

Results show ：

7.1 checkNNGradients.m

function checkNNGradients(lambda)
%CHECKNNGRADIENTS Creates a small neural network to check the
%backpropagation gradients
%   CHECKNNGRADIENTS(lambda) Creates a small neural network to check the
%   backpropagation gradients, it will output the analytical gradients
%   produced by your backprop code and the numerical gradients (computed
%   using computeNumericalGradient). These two gradient computations should
%   result in very similar values.
%

if ~exist('lambda', 'var') || isempty(lambda)
    lambda = 0;
end

input_layer_size = 3;
hidden_layer_size = 5;
num_labels = 3;
m = 5;

% We generate some 'random' test data
Theta1 = debugInitializeWeights(hidden_layer_size, input_layer_size);
Theta2 = debugInitializeWeights(num_labels, hidden_layer_size);
% Reusing debugInitializeWeights to generate X
X  = debugInitializeWeights(m, input_layer_size - 1);
y  = 1 + mod(1:m, num_labels)'; % Unroll parameters nn_params = [Theta1(:) ; Theta2(:)]; % Short hand for cost function costFunc = @(p) nnCostFunction(p, input_layer_size, hidden_layer_size, ... num_labels, X, y, lambda); [cost, grad] = costFunc(nn_params); numgrad = computeNumericalGradient(costFunc, nn_params); % Visually examine the two gradient computations. The two columns % you get should be very similar. disp([numgrad grad]); fprintf(['The above two columns you get should be very similar.\n' ... '(Left-Your Numerical Gradient, Right-Analytical Gradient)\n\n']); % Evaluate the norm of the difference between two solutions. % If you have a correct implementation, and assuming you used EPSILON = 0.0001 % in computeNumericalGradient.m, then diff below should be less than 1e-9 diff = norm(numgrad-grad)/norm(numgrad+grad); fprintf(['If your backpropagation implementation is correct, then \n' ... 'the relative difference will be small (less than 1e-9). \n' ... '\nRelative Difference: %g\n'], diff);

end

8. Regularization implementation of back propagation algorithm

%% =============== Part 8: Implement Regularization ===============
%  Once your backpropagation implementation is correct, you should now
%  continue to implement the regularization with the cost and gradient.
%

fprintf('\nChecking Backpropagation (w/ Regularization) ... \n')

%  Check gradients by running checkNNGradients
lambda = 3;
checkNNGradients(lambda);

% Also output the costFunction debugging values
debug_J  = nnCostFunction(nn_params, input_layer_size, ...
                          hidden_layer_size, num_labels, X, y, lambda);

fprintf(['\n\nCost at (fixed) debugging parameters (w/ lambda = %f): %f ' ...
         '\n(for lambda = 3, this value should be about 0.576051)\n\n'], lambda, debug_J);

fprintf('Program paused. Press enter to continue.\n');

Results show ：

It can be seen from the above figure , This is a process that involves a lot of computation , It's going to be a lot of trouble .

9. Neural network training

%% =================== Part 8: Training NN ===================
%  You have now implemented all the code necessary to train a neural
%  network. To train your neural network, we will now use "fmincg", which
%  is a function which works similarly to "fminunc". Recall that these
%  advanced optimizers are able to train our cost functions efficiently as
%  long as we provide them with the gradient computations.
%
fprintf('\nTraining Neural Network... \n')

%  After you have completed the assignment, change the MaxIter to a larger
%  value to see how more training helps.
options = optimset('MaxIter', 50);

%  You should also try different values of lambda
lambda = 1;

% Create "short hand" for the cost function to be minimized
costFunction = @(p) nnCostFunction(p, ...
                                   input_layer_size, ...
                                   hidden_layer_size, ...
                                   num_labels, X, y, lambda);

% Now, costFunction is a function that takes in only one argument (the
% neural network parameters)
[nn_params, cost] = fmincg(costFunction, initial_nn_params, options);

% Obtain Theta1 and Theta2 back from nn_params
Theta1 = reshape(nn_params(1:hidden_layer_size * (input_layer_size + 1)), ...
                 hidden_layer_size, (input_layer_size + 1));

Theta2 = reshape(nn_params((1 + (hidden_layer_size * (input_layer_size + 1))):end), ...
                 num_labels, (hidden_layer_size + 1));

fprintf('Program paused. Press enter to continue.\n');

This process is time-consuming

Results are for reference only ：

10. Weight and prediction accuracy

%% ================= Part 9: Visualize Weights =================
%  You can now "visualize" what the neural network is learning by
%  displaying the hidden units to see what features they are capturing in
%  the data.

fprintf('\nVisualizing Neural Network... \n')

displayData(Theta1(:, 2:end));

fprintf('\nProgram paused. Press enter to continue.\n');
pause;

%% ================= Part 10: Implement Predict =================
%  After training the neural network, we would like to use it to predict
%  the labels. You will now implement the "predict" function to use the
%  neural network to predict the labels of the training set. This lets
%  you compute the training set accuracy.

pred = predict(Theta1, Theta2, X);

fprintf('\nTraining Set Accuracy: %f\n', mean(double(pred == y)) * 100);

10.1 predict.m

function p = predict(Theta1, Theta2, X)
%PREDICT Predict the label of an input given a trained neural network
%   p = PREDICT(Theta1, Theta2, X) outputs the predicted label of X given the
%   trained weights of a neural network (Theta1, Theta2)

% Useful values
m = size(X, 1);
num_labels = size(Theta2, 1);

% You need to return the following variables correctly 
p = zeros(size(X, 1), 1);

h1 = sigmoid([ones(m, 1) X] * Theta1'); h2 = sigmoid([ones(m, 1) h1] * Theta2');
[dummy, p] = max(h2, [], 2);

% =========================================================================


end

The above is the whole process of handwritten digit recognition , You only need to complete the top three questions , Understand the code again , Can complete NG Your homework . Of course , I have the code for the training part of the neural network , Not very proficient at it , Need to continue to strengthen the practice .