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稀疏自编码
2016-12-26 10:07:23       个评论    来源:dataningwei的博客  
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稀疏编码代价函数J ,给定一个包含M个样例的数据集,代价函数为:

\

其中λ为权重衰减因子,其目的是要减小权重的幅度,防止过拟合。β控制稀疏惩罚因子的权重。

具体原理推导可参考UFLDL Autoencoders And Sparsity教程。代价函数及梯度计算代码如下

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. 
% 
% data = [ones(1,size(data,2)); data];  
[n m] = size(data);  

%前向计算
z2 = W1* data + repmat(b1,1,m);  
a2 = sigmoid(z2);  
z3 = W2 * a2 + repmat(b2,1,m);   
a3 = sigmoid(z3);  

%计算隐含层平均活跃度 
phat = mean(a2,2);  
p = repmat(sparsityParam, size(phat));  
sparse = p .* log(p ./ phat) + (1-p) .* log((1-p) ./ (1-phat));  

J = sum(sum((a3-data).^2)) / (m*2);  

regu = (W1(:)'*W1(:) + W2(:)'*W2(:))/2;  
%代价函数
cost = J + lambda*regu + beta * sum(sparse);  

delta3 = -1* (data-a3).*a3.*(1-a3);  
delta2 = (W2'*delta3+beta*repmat(-p./phat+(1-p)./(1-phat),1,size(data,2))).*a2.*(1-a2);   

W2grad = delta3*a2'/m;    
b2grad = mean(delta3,2);    
W1grad = delta2*data'/m;    
b1grad = mean(delta2,2);   

W2grad = W2grad + lambda*W2;    
W1grad = W1grad + lambda*W1;

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