本实验是用线性解码器的sparse autoencoder来训练stl-10数据库图片中8*8大小RGB patch块的特征。之前的试验中，我们的训练图像都是灰度图像，对于RGB图像可以采用相同的方法，只需把RGB图像的三个通道向量按照rgb的顺序排列更长的向量即可。
1、 线性解码器简介
线性解码器和 稀疏自动编码的整体结构都是类似的，只是线性解码器的输出层的激励函数为 恒等式f(z) = z，而稀疏自动编码为 非线性函数，比如sigmoid函数、tanh函数等，那为什么为出现两者不同的激励函数呢。我们以三层神经网络为例，输出层的计算公式如下：
其中a(3)是第二层的输出，在自编码器中,我们希望a(3)近似重构了输入x=a(1)。
我们在稀疏自动编码采用sigmoid激励函数，把输出数据控制在[0，1]范围，如果输入数据能够方便缩放到 [0,1] 中（MNIST手写数字数据集），那稀疏自动编码可以满足要求。但是，有些数据很难把数据缩放到[0,1]之间（比如PCA 白化处理的输入），如果此时输入数据的范围为[0,10]，仍然使用sigmoid激励函数保证输出为[0,1]，此时不能得到输出重构输入的要求，在这种情况下，线性解码器的优势就体现出来了。
2、 神经网络结构
实验中为三层神经网络，输入层8*8*3个neuron，隐含层为400个neuron（都不包括bias结点），输出层为8*8*3个neuron。
3、 数据
实验中的数据采用的100,000个8 x 8的RGB patch块，这些patch是从STL-10图像集中随机采样得到的。STL-10数据中由5000个训练数据和8000个训练数据组成，每一个数据是大小为96x96标注的彩色图像，这数据属于airplane, bird, car, cat, deer, dog, horse, monkey, ship, truck十个类中的一类。
4、 预处理（ZCA白化）
本实验中采用来预处理，降低输入数据的冗余性，从而降低特征之间相关性，以及使所有特征具有相同的方差。
下图是把训练数据中的前100个patch进行显示。
原始数据 ZCA白化处理后
5、 实验结果
下图是通过线性解码器从STL-10数据集中学习到的特征。
6、 代码
源代码下载sparseAutoencoderLinearCost.m文件
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.
%
% data = [ones(1,size(data,2)); data];
[n m] = size(data);
z2 = W1* data + repmat(b1,1,m);
a2 = sigmoid(z2);
z3 = W2 * a2 + repmat(b2,1,m);
a3 = z3;
%J = trace((data - a3)' * (data - a3)) / (m*2);
J = sum(sum((a3-data).^2)) / (m*2);
regu = (W1(:)'*W1(:) + W2(:)'*W2(:))/2;
phat = mean(a2,2);
p = repmat(sparsityParam, size(phat));
sparse = p .* log(p ./ phat) + (1-p) .* log((1-p) ./ (1-phat));
cost = J + lambda*regu + beta * sum(sparse);
delta3 = -1* (data-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