稀疏自编码器(Sparse Autoencoder)可以自动从无标注数据中学习特征，可以给出比原始数据更好的特征描述。在实际运用时可以用稀疏编码器发现的特征取代原始数据，这样往往能带来更好的结果。本文将给出稀疏自编码器的算法描述，并演示说明稀疏编码器自动提取边缘特征。
下面，给出稀疏自编码器代价函数及其导数的matlab代码实现：
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.
%
m=size(data,2);
x=data;
a1=x;
z2=W1*a1+repmat(b1,1,m);
a2=sigmoid(z2);
z3=W2*a2+repmat(b2,1,m);
a3=sigmoid(z3);
h=a3;
y=x;
squared_error=0.5*sum((h-y).^2,1);
rho=1/m*sum(a2,2);
sparsity_penalty= beta*sum(sparsityParam.*log(sparsityParam./rho)+(1-sparsityParam).*log((1-sparsityParam)./(1-rho)));
cost=1/m*sum(squared_error)+lambda/2*(sum(sum(W1.^2))+sum(sum(W2.^2))) + sparsity_penalty;
grad_z3=a3.*(1-a3);
delta_3=-(y-a3).*grad_z3;
grad_z2=a2.*(1-a2);
delta_2=(W2'*delta_3+repmat(beta*(-sparsityParam./rho+(1-sparsityParam)./(1-rho)),1,m)).*grad_z2;
Delta_W2=delta_3*a2';
Delta_b2=sum(delta_3,2);
Delta_W1=delta_2*a1';
Delta_b1=sum(delta_2,2);
W1grad=1/m*Delta_W1+lambda*W1;
W2grad=1/m*Delta_W2+lambda*W2;
b1grad=1/m*Delta_b1;
b2grad=1/m*Delta_b2;
%-------------------------------------------------------------------
% 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
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