Exercise:Learning color features with Sparse Autoencoders

参考网页：

　　http://deeplearning.stanford.edu/wiki/index.php/Linear_Decoders

　　http://deeplearning.stanford.edu/wiki/index.php/Exercise:Learning_color_features_with_Sparse_Autoencoders

实验介绍：彩色图片的特征学习，采用STL-10，包含100,000个大小为8*8的patch 块。由于参数求解过程中对内存需要比较大，作者仅使得5000个patch块进行实验。

　　　　环境：matlab 2010a

实验模型：

　　Linear Decoder模型，与三层的autoencoders模型类似，不同之处在于输出层的激活函数为线性激活函数，即f(z) = z 。　

理论基础：

　　Sparse Autoencoder Recap: 在稀疏自编码中，我们使用三层神经网络：输出层，隐含层，输出层。在隐含层、输出层我们使用相同的激活函数（sigmoid）。

在输出层a表示输出，在autoencoder中，a3表示对输入x的重构。由于在该模型中使用sigmoid激活函数，它的输出在[0 , 1]之间，因此我们需要将输入缩放到[0 , 1]之间。这种对输出的缩放模型对于MNIST数据集非常适合，但他对其它的应用场合并非都适合。比如，我们对输入样本执行PCA 白化，那么输入不再属于[0 , 1]，因此autoencoder不能解决该问题。

　　Linear Decoder:结果以上问题的简单的方法就是在输出层使：a⁽³⁾ = z⁽³⁾，即在输出层的激活函数为identity function f(z) = z 。这种激活函数称为：线性激活函数。这样输出a⁽³⁾可以是任意值，没有[0 , 1]的限制，有篮球稀疏自编码模型的扩展。

　　　　因此在该模型中同样包含三层神经网络：输入层，隐含层，输出层。隐含层使用sigmoid激活函数，输出层使用线性激活函数。由于输出层的激活函数发生变化，因此梯度计算模块也随之发生变化。

实验步骤：

　　0、以前实验中都使用灰度图片，在该实验中我们使用彩色图片，即包含RGB三个通道，输入即把三个通道值串联起来，那么输入的大小为8*8*3

　　1、简单使用Linear Decoder模型进行特征学习

　　2、首先对输入作ZCA whitening，然后再使用白化后的结果作为Linear Decoder的输入进行特征学习。

实验结果：

　　原始样本：

　　

　　白化：

　　

　　权重可视化：

　　

　　可以看到学习到的特征为一些边缘。

实验主要代码：

　sparseAutoencoderLinearCost.m

注意：作者给的源码中，该函数的返回值有三个[const,grad ,features]，在调用的时候仅使用前两个值，把features去掉即可，

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. 
% 

Jcost = 0;%
Jweight = 0;%
Jsparse = 0;%
[n m] = size(data);%mn

%修改部分
z2 = W1*data+repmat(b1,1,m);%b1m
a2 = sigmoid(z2);
z3 = W2*a2+repmat(b2,1,m);
a3 = z3;

% 
Jcost = (0.5/m)*sum(sum((a3-data).^2));

%
Jweight = (1/2)*(sum(sum(W1.^2))+sum(sum(W2.^2)));

%
rho = (1/m).*sum(a2,2);%
Jsparse = sum(sparsityParam.*log(sparsityParam./rho)+ ...
        (1-sparsityParam).*log((1-sparsityParam)./(1-rho)));

%
cost = Jcost+lambda*Jweight+beta*Jsparse;

%修改部分
d3 = -(data-a3);
sterm = beta*(-sparsityParam./rho+(1-sparsityParam)./(1-rho));%
                                                             %
d2 = (W2'*d3+repmat(sterm,1,m)).*sigmoidInv(z2); 

%W1grad 
W1grad = W1grad+d2*data';
W1grad = (1/m)*W1grad+lambda*W1;

%W2grad  
W2grad = W2grad+d3*a2';
W2grad = (1/m).*W2grad+lambda*W2;

%b1grad 
b1grad = b1grad+sum(d2,2);
b1grad = (1/m)*b1grad;%b

%b2grad 
b2grad = b2grad+sum(d3,2);
b2grad = (1/m)*b2grad;




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

%sigmoid
function sigmInv = sigmoidInv(x)

    sigmInv = sigmoid(x).*(1-sigmoid(x));
end