1、Multi-class Classification

如果將這個題轉換為神經網路，相當於這個模型只有兩層：輸入層和輸出層，輸入層由400個神經元（畫素）組成，輸出層由10個神經元組成，輸出層的神經元編號為1到10，分別表示1到9和0（10表示0），每個神經元輸出結果是預測輸入影象是該神經元編號的概率，選取概率最大的神經元編號作為預測的數字。

1.3 Vectorizing Logistic Regression

function [J, grad] = lrCostFunction(theta, X, y, lambda)

% Initialize some useful values
m = length(y); % number of training examples

% You need to return the following variables correctly 
J = 0;
grad = zeros(size(theta));

J = (-y' * log(sigmoid(X * theta)) - (1 - y)' * log(1 - sigmoid(X * theta))) / m ...
		+ lambda / 2 / m * sum(theta(2 : end) .^ 2);

temp = theta;
temp(1) = 0;
grad = (X' * (sigmoid(X * theta) - y) + lambda * temp) / m;


grad = grad(:);

end
 

1.4 One-vs-all Classification

注意呼叫fmincg時，theta的初始值為 all_theta(c,:)' ，有轉置。

function [all_theta] = oneVsAll(X, y, num_labels, lambda)

% Some useful variables
m = size(X, 1);		% 行數 5000
n = size(X, 2);		% 列數 400

% You need to return the following variables correctly 
all_theta = zeros(num_labels, n + 1);	% 10 * 401

% Add ones to the X data matrix
X = [ones(m, 1) X];		% 5000 * 401

cost = 0;
options = optimset('GradObj', 'on', 'MaxIter', 50);

for c = 1 : num_labels
	% 傳入 lrCostFunction() 裡的 theta 是列向量，所以 all_theta(c,:)'
	all_theta(c,:) = fmincg(@(t)(lrCostFunction(t, X, (y == c), lambda)), all_theta(c,:)', options)';
endfor


end
 

1.4.1 One-vs-all Prediction

function p = predictOneVsAll(all_theta, X)


m = size(X, 1);
num_labels = size(all_theta, 1);

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

% Add ones to the X data matrix
X = [ones(m, 1) X];
    


% X: 5000 * 401, all_theta: 10 * 401
% X * all_theta': 5000 * 10
[maxx, p] = max(X * all_theta', [], 2);


end
 

2 Neural Networks

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);


X = [ones(m, 1) X];
a2 = [ones(m, 1) sigmoid(X * Theta1')];
%size(a2)
[maxx, p] = max(sigmoid(a2 * Theta2'), [], 2);


end