UFLDL Exercise:Learning color features with Sparse Autoencoders

这一节的内容比较简单，就是实现一个线性解码器，为什么要什么用线性呢，因为在有些应用的场景（如用pca白化处理的数据，因为数据的均值为零，方差为1，所以不一定能落在0~1的范围内）里，输入是不能缩放到0~1之间的，而s型激励函数的输出是0~1，所以我们就只能线性函数来作为输出层的激励函数了

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层输出
a2 = bsxfun(@plus,W2 * a1,b2); %输出层输出，为恒等激励
p = mean(a1,2); %隐藏神经元的平均活跃度
sparsity = sparsityParam .* log(sparsityParam ./ p) + (1 - sparsityParam) .* log((1 - sparsityParam) ./ (1.-p)); %惩罚因子
%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); %代价函数
delt2 = (a2 - data); %输出层残差，注意这里用的是恒等激励，所以导数为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层残差
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

当然，在写完sparseAutoencoderLinearCost的代码后还是要check gradient，保证代码没问题才进行下一步~
接下来只需执行它提供的代码就可以看到自编码器的学到了什么~