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. 

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. 

% 

% input data

x=data;

% for autoencoder output y is equal to x

y=x; 

% length of training data

m=size(x,2);

% Vectorized implementation of forward propagation

z2 = W1 * x + repmat(b1,1,m);

a2 = sigmoid(z2);

z3 = W2 * a2 + repmat(b2,1,m);

h = sigmoid(z3);

% squared error term

hmy=h-y;

J_sq=sum(sum(hmy.*hmy))/(2*m);

% weight decay term

J_wd=lambda/(2)*(trace(W1'*W1)+trace(W2'*W2));

% sparsity penalty

rho=sparsityParam;

rho_hat=mean(a2,2);

KL=rho.*log(rho./rho_hat)+(1-rho).*log((1-rho)./(1-rho_hat));

J_sp=beta*sum(KL);

% cost= J_[squared_error]+J_[weight_decay]+J_[sparsity_penalty]

cost=J_sq+J_wd+J_sp;

% gradient

a1=x;

delta3 = -(y - h) .* fprime(z3); 

sparsity_delta = - rho ./ rho_hat + (1 - rho) ./ (1 - rho_hat);

sd_mat=repmat(sparsity_delta,1,m);

delta2 = (W2'*delta3+beta*sd_mat) .* fprime(z2);

W2grad =  delta3*a2'/m+lambda*W2;

W1grad =  delta2*a1'/m+lambda*W1; 

b2grad = delta3 * ones(m,1)/m;

b1grad = delta2* ones(m,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

% derivative of sigmoid function

% also we can implement it using: f'(x)= f(x)(1-f(x)) if f(x)=sigmoid

function fp = fprime(x)

    fp = exp(-x) ./ (1 + exp(-x)).^2;

end