UFLDL Exercise:Learning color features with Sparse Autoencoders

這一節的內容比較簡單，就是實現一個線性解碼器，爲什麼要什麼用線性呢，因爲在有些應用的場景（如用pca白化處理的數據，因爲數據的均值爲零，方差爲1，所以不一定能落在0~1的範圍內）裏，輸入是不能縮放到0~1之間的，而s型激勵函數的輸出是0~1，所以我們就只能線性函數來作爲輸出層的激勵函數了

STEP 1: Create and modify sparseAutoencoderLinearCost.m to use a linear decoder

sparseAutoencoderLinearCost.m

function [cost,grad] = sparseAutoencoderLinearCost(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. 
% 
a1 = sigmoid(bsxfun(@plus,W1 * data,b1)); %hidden層輸出
a2 = bsxfun(@plus,W2 * a1,b2); %輸出層輸出，爲恆等激勵
p = mean(a1,2); %隱藏神經元的平均活躍度
sparsity = sparsityParam .* log(sparsityParam ./ p) + (1 - sparsityParam) .* log((1 - sparsityParam) ./ (1.-p)); %懲罰因子
%cost = sum(sum((a2 - data).^2)) / 2 / size(data,2);
%cost = sum(sum((a2 - data).^2)) / 2 / size(data,2) + lambda / 2 * (sum(sum(W1.^2)) + sum(sum(W2.^2)));
%cost = sum(sum((a2 - data).^2)) / 2 / size(data,2) + beta * sum(sparsity);
cost = sum(sum((a2 - data).^2)) / 2 / size(data,2) + lambda / 2 * (sum(sum(W1.^2)) + sum(sum(W2.^2))) + beta * sum(sparsity); %代價函數
delt2 = (a2 - data); %輸出層殘差，注意這裏用的是恆等激勵，所以導數爲1
%delt1 = W2' * delt2 .* a1 .* (1 - a1);
delt1 = (W2' * delt2 + beta .* repmat((-sparsityParam./p + (1-sparsityParam)./(1.-p)),1,size(data,2))) .* a1 .* (1 - a1); %hidden層殘差
W2grad = delt2 * a1' ./ size(data,2) + lambda * W2; 
W1grad = delt1 * data' ./ size(data,2) + lambda * W1;
% W2grad = delt2 * a1' ./ size(data,2);
% W1grad = delt1 * data' ./ size(data,2);
b2grad = sum(delt2,2) ./ size(data,2);
b1grad = sum(delt1,2) ./ size(data,2);

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

當然，在寫完sparseAutoencoderLinearCost的代碼後還是要check gradient，保證代碼沒問題才進行下一步~

接下來只需執行它提供的代碼就可以看到自編碼器的學到了什麼~