sparseAutoencoderCost.m

function [cost,grad] = sparseAutoencoderCost(theta, visibleSize, hiddenSize, ...
                                             lambda, sparsityParam, beta, data)

                                        
                                  
                                        
                                         
% visibleSize: the number of input units (probably 64) 
% hiddenSize: the number of hidden units (probably 25) 
% lambda: weight decay parameter
% sparsityParam: The desired average activation for the hidden units (denoted in the lecture
%                           notes by the greek alphabet rho, which looks like a lower-case "p").
% beta: weight of sparsity penalty term
% data: Our 64x10000 matrix containing the training data.  So, data(:,i) is the i-th training example. 
  
% The input theta is a vector (because minFunc expects the parameters to be a vector). 
% We first convert theta to the (W1, W2, b1, b2) matrix/vector format, so that this 
% follows the notation convention of the lecture notes. 

W1 = reshape(theta(1:hiddenSize*visibleSize), hiddenSize, visibleSize);
W2 = reshape(theta(hiddenSize*visibleSize+1:2*hiddenSize*visibleSize), visibleSize, hiddenSize);
b1 = theta(2*hiddenSize*visibleSize+1:2*hiddenSize*visibleSize+hiddenSize);
b2 = theta(2*hiddenSize*visibleSize+hiddenSize+1:end);

% Cost and gradient variables (your code needs to compute these values). 
% Here, we initialize them to zeros. 

% just for test


cost = 0;
W1grad = zeros(size(W1)); 
W2grad = zeros(size(W2));
b1grad = zeros(size(b1)); 
b2grad = zeros(size(b2));

%% ---------- YOUR CODE HERE --------------------------------------
%  Instructions: Compute the cost/optimization objective J_sparse(W,b) for the Sparse Autoencoder,
%                and the corresponding gradients W1grad, W2grad, b1grad, b2grad.
%
% W1grad, W2grad, b1grad and b2grad should be computed using backpropagation.
% Note that W1grad has the same dimensions as W1, b1grad has the same dimensions
% as b1, etc.  Your code should set W1grad to be the partial derivative of J_sparse(W,b) with
% respect to W1.  I.e., W1grad(i,j) should be the partial derivative of J_sparse(W,b) 
% with respect to the input parameter W1(i,j).  Thus, W1grad should be equal to the term 
% [(1/m) \Delta W^{(1)} + \lambda W^{(1)}] in the last block of pseudo-code in Section 2.2 
% of the lecture notes (and similarly for W2grad, b1grad, b2grad).
% 
% Stated differently, if we were using batch gradient descent to optimize the parameters,
% the gradient descent update to W1 would be W1 := W1 - alpha * W1grad, and similarly for W2, b1, b2. 
% 

[n,m] = size(data);


%% activation value of the one iteration
z2 = W1 * data + repmat(b1,1,m);
a2 = sigmoid(z2);
z3 = W2 * a2 + repmat(b2,1,m);
a3 = sigmoid(z3);

sums = (data - a3).^2;

p_hat = sum(a2,2).*(1/m);


%% J(W,b,x,y)
W1_square = W1.^2;
W2_square = W2.^2;

kl = KL(sparsityParam,p_hat);

cost = 0.5 * sum(sums(:)) * (1/m) + 0.5 * lambda * (  sum(W1_square(:)) + sum(W2_square(:)) ) + beta * sum(kl);
      
%%

err_out = -1 * (data - a3).*a3.*(1 - a3);

err_hidden = ((W2' * err_out)  ...,
 + repmat(beta * (-sparsityParam./p_hat + (1-sparsityParam)./(1-p_hat)),1,m)) ...,
 .*(a2).*(1-a2);

W1grad = (err_hidden * data').*(1/m)  + lambda.*W1;
W2grad = (err_out * a2' ).* (1/m) + lambda.*W2 ;

b1grad = sum(err_hidden,2).*(1/m);
b2grad = sum(err_out,2).*(1/m);

%-------------------------------------------------------------------
% After computing the cost and gradient, we will convert the gradients back
% to a vector format (suitable for minFunc).  Specifically, we will unroll
% your gradient matrices into a vector.

grad = [W1grad(:) ; W2grad(:) ; b1grad(:) ; b2grad(:)];

end

%-------------------------------------------------------------------
% Here's an implementation of the sigmoid function, which you may find useful
% in your computation of the costs and the gradients.  This inputs a (row or
% column) vector (say (z1, z2, z3)) and returns (f(z1), f(z2), f(z3)). 

function sigm = sigmoid(x)
  
    sigm = 1 ./ (1 + exp(-x));
end


%% here is the kl divergence function
function kl = KL(p,p1)

    kl = p.*log(p./p1) + (1-p).*log((1-p)./(1-p1) );

end