前言
实验内容：Exercise:Learning color features with Sparse Autoencoders。即：利用线性解码器，从100000张8*8的RGB图像块中提取颜色特征，这些特征会被用于下一节的练习
理论知识：线性解码器和http://www.cnblogs.com/tornadomeet/archive/2013/04/08/3007435.html
实验基础说明：
1.为什么要用线性解码器，而不用前面用过的栈式自编码器等？即：线性解码器的作用？
这一点，Ng已经在讲解中说明了，因为线性解码器不用要求输入数据范围一定为(0,1），而前面用过的栈式自编码器等要求输入数据范围必须为(0,1）。因为a3的输出值是f函数的输出，而在普通的sparse autoencoder中f函数一般为sigmoid函数，所以其输出值的范围为(0,1)，所以可以知道a3的输出值范围也在0到1之间。另外我们知道，在稀疏模型中的输出层应该是尽量和输入层特征相同，也就是说a3=x1，这样就可以推导出x1也是在0和1之间，那就是要求我们对输入到网络中的数据要先变换到0和1之间，这一条件虽然在有些领域满足，比如前面实验中的MINIST数字识别。但是有些领域，比如说使用了PCA Whitening后的数据，其范围却不一定在0和1之间。因此Linear Decoder方法就出现了。Linear Decoder是指在隐含层采用的激发函数是sigmoid函数，而在输出层的激发函数采用的是线性函数，比如说最特别的线性函数——等值函数。
2.在实验中，在ZCA whitening前进行数据预处理时，每列代表一个样本，但为什么是对patches的每行0均值化（即：每一维度0均值化，具体做法是：首先计算每一个维度上数据的均值（使用全体数据计算），之后在每一个维度上都减去该均值。），而以前的实验都是对每列即每个样本0均值化（即：逐样本均值消减）？
①因为以前是灰度图，现在是RGB彩色图像，如果现在对每列平均就是对三个通道求平均，这肯定不行。因为不同色彩通道中的像素并不都存在平稳特性，而要进行逐样本均值消减（即：单独每个样本0均值化）有一个必须满足的前提：该数据是平稳的（见：数据预处理）。
平稳性的理解可见：http://lidequan12345.blog.163.com/blog/static/28985036201177892790。
②因为以前是自然图像，自然图像中像素之间的统计特性都一样，有一定的相关性，而现在是人工分割的图像块，没有这种特性。
3.在实验中，把网络权值显示出来为什么是用displayColorNetwork( (W*ZCAWhite)’)，而不像以前用的是display_Network( (W1)’)?
因为在本实验中，数据patches在输入网络前先经过了ZCA whitening的数据预处理，变成了ZCA白化后的数据ZCAWhite * patches，所以第一层隐含层输出的实际上是W*ZCAWhite * patches，也就是说从原始数据patches到第一层隐含层输出为W*ZCAWhite * patches的整个过程l转换权值为W*ZCAWhite。
4.PCA Whitening和ZCA Whitening的区别？即：为什么本实验没用PCA Whitening
PCA Whitening：处理后的各数据方差都都相等，并都为1。主要用于降维和去相关性。
ZCA Whitening：处理后的各数据方差不一定为1，但一定相等。主要用于去相关性，且能尽量保持原始数据。
5.优秀的编程技巧：
要学会用函数句柄，比如patches = bsxfun(@minus, patches, meanPatch);
因为不使用函数句柄的情况下，对函数多次调用，每次都要为该函数进行全面的路径搜索，直接影响计算速度，借助句柄可以完全避免这种时间损耗。也就是直接指定了函数的指针。函数句柄就像一个函数的名字，有点类似于C++程序中的引用。当然这一点已经在Deep Learning一之深度学习UFLDL教程：Sparse Autoencoder练习（斯坦福大学深度学习教程）中提到过，但我觉得有必须再强调一下。
实验步骤
1.初始化参数，编写计算线性解码器代价函数及其梯度的函数sparseAutoencoderLinearCost.m，主要是在sparseAutoencoderCost.m的基础上稍微修改，然后再检查其梯度实现是否正确。
2.加载数据并原始数据进行ZCA Whitening的预处理。
3.学习特征，即用LBFG算法训练整个线性解码器网络，得到整个网络权值optTheta。
4.可视化第一层学习到的特征。
实验结果
原始数据
ZCA Whitening后的数据
特征可视化结果，即：每一层学习到的特征
代码
linearDecoderExercise.m
%% CS294A/CS294W Linear Decoder Exercise% Instructions
% ------------
%
% This file contains code that helps you get started on the
% linear decoder exericse. For this exercise, you will only need to modify
% the code in sparseAutoencoderLinearCost.m. You will not need to modify
% any code in this file.%%======================================================================
%% STEP : Initialization
% Here we initialize some parameters used for the exercise.imageChannels = ; % number of channels (rgb, so )patchDim = ; % patch dimension
numPatches = ; % number of patchesvisibleSize = patchDim * patchDim * imageChannels; % number of input units
outputSize = visibleSize; % number of output units
hiddenSize = ; % number of hidden units sparsityParam = 0.035; % desired average activation of the hidden units.
lambda = 3e-; % weight decay parameter
beta = ; % weight of sparsity penalty term epsilon = 0.1; % epsilon for ZCA whitening%%======================================================================
%% STEP : Create and modify sparseAutoencoderLinearCost.m to use a linear decoder,
% and check gradients
% You should copy sparseAutoencoderCost.m from your earlier exercise
% and rename it to sparseAutoencoderLinearCost.m.
% Then you need to rename the function from sparseAutoencoderCost to
% sparseAutoencoderLinearCost, and modify it so that the sparse autoencoder
% uses a linear decoder instead. Once that is done, you should check
% your gradients to verify that they are correct.% NOTE: Modify sparseAutoencoderCost first!% To speed up gradient checking, we will use a reduced network and some
% dummy patchesdebugHiddenSize = ;
debugvisibleSize = ;
patches = rand([ ]);
theta = initializeParameters(debugHiddenSize, debugvisibleSize); [cost, grad] = sparseAutoencoderLinearCost(theta, debugvisibleSize, debugHiddenSize, ...
lambda, sparsityParam, beta, ...
patches);% Check gradients
numGrad = computeNumericalGradient( @(x) sparseAutoencoderLinearCost(x, debugvisibleSize, debugHiddenSize, ...
lambda, sparsityParam, beta, ...
patches), theta);% Use this to visually compare the gradients side by side
disp([numGrad grad]); diff = norm(numGrad-grad)/norm(numGrad+grad);
% Should be small. In our implementation, these values are usually less than 1e-.
disp(diff); assert(diff < 1e-, 'Difference too large. Check your gradient computation again');% NOTE: Once your gradients check out, you should run step again to
% reinitialize the parameters
%}%%======================================================================
%% STEP : 从pathes中学习特征 Learn features on small patches
% In this step, you will use your sparse autoencoder (which now uses a
% linear decoder) to learn features on small patches sampled from related
% images.%% STEP 2a: 加载数据 Load patches
% In this step, we load 100k patches sampled from the STL10 dataset and
% visualize them. Note that these patches have been scaled to [,]load stlSampledPatches.mat %怎么就就这个变量加到pathes上了呢？因为它里面自己定义了变量patches的值！
figure;
displayColorNetwork(patches(:, :)); %% STEP 2b: 预处理 Apply preprocessing
% In this sub-step, we preprocess the sampled patches, in particular,
% ZCA whitening them.
%
% In a later exercise on convolution and pooling, you will need to replicate
% exactly the preprocessing steps you apply to these patches before
% using the autoencoder to learn features on them. Hence, we will save the
% ZCA whitening and mean image matrices together with the learned features
% later on.% Subtract mean patch (hence zeroing the mean of the patches)
meanPatch = mean(patches, ); %为什么是对每行求平均，以前是对每列即每个样本求平均呀？因为以前是灰度图，现在是彩色图，如果现在对每列平均就是对三个通道求平均，这肯定不行
patches = bsxfun(@minus, patches, meanPatch);% Apply ZCA whitening
sigma = patches * patches' / numPatches; %协方差矩阵
[u, s, v] = svd(sigma);
ZCAWhite = u * diag( ./ sqrt(diag(s) + epsilon)) * u';
patches = ZCAWhite * patches;figure;
displayColorNetwork(patches(:, :));%% STEP 2c: Learn features
% You will now use your sparse autoencoder (with linear decoder) to learn
% features on the preprocessed patches. This should take around minutes.theta = initializeParameters(hiddenSize, visibleSize);% Use minFunc to minimize the function
addpath minFunc/options = struct;
options.Method = 'lbfgs';
options.maxIter = ;
options.display = 'on';[optTheta, cost] = minFunc( @(p) sparseAutoencoderLinearCost(p, ...
visibleSize, hiddenSize, ...
lambda, sparsityParam, ...
beta, patches), ...
theta, options);% Save the learned features and the preprocessing matrices for use in
% the later exercise on convolution and pooling
fprintf('Saving learned features and preprocessing matrices...\n');
save('STL10Features.mat', 'optTheta', 'ZCAWhite', 'meanPatch');
fprintf('Saved\n');%% STEP 2d: Visualize learned featuresW = reshape(optTheta(:visibleSize * hiddenSize), hiddenSize, visibleSize);
b = optTheta(*hiddenSize*visibleSize+:*hiddenSize*visibleSize+hiddenSize);
figure;
displayColorNetwork( (W*ZCAWhite)');
sparseAutoencoderLinearCost.m
function [cost,grad,features] = sparseAutoencoderLinearCost(theta, visibleSize, hiddenSize, ...
lambda, sparsityParam, beta, data)
%计算线性解码器代价函数及其梯度
% visibleSize:输入层神经单元节点数
% hiddenSize:隐藏层神经单元节点数
% lambda: 权重衰减系数
% sparsityParam: 稀疏性参数
% beta: 稀疏惩罚项的权重
% data: 训练集
% theta：参数向量，包含W1、W2、b1、b2
% -------------------- YOUR CODE HERE --------------------
% Instructions:
% Copy sparseAutoencoderCost in sparseAutoencoderCost.m from your
% earlier exercise onto this file, renaming the function to
% sparseAutoencoderLinearCost, and changing the autoencoder to use a
% linear decoder.
% -------------------- YOUR CODE HERE --------------------
% The input theta is a vector because minFunc only deal with vectors. In
% this step, we will convert theta to matrix format such that they follow
% the notation in the lecture notes.
W1 = reshape(theta(:hiddenSize*visibleSize), hiddenSize, visibleSize);
W2 = reshape(theta(hiddenSize*visibleSize+:*hiddenSize*visibleSize), visibleSize, hiddenSize);
b1 = theta(*hiddenSize*visibleSize+:*hiddenSize*visibleSize+hiddenSize);
b2 = theta(*hiddenSize*visibleSize+hiddenSize+:end);% Loss and gradient variables (your code needs to compute these values)
m = size(data, ); % 样本数量%% ---------- YOUR CODE HERE --------------------------------------
% Instructions: Compute the loss for the Sparse Autoencoder and gradients
% W1grad, W2grad, b1grad, b2grad
%
% Hint: ) data(:,i) is the i-th example
% ) your computation of loss and gradients should match the size
% above for loss, W1grad, W2grad, b1grad, b2grad% z2 = W1 * x + b1
% a2 = f(z2)
% z3 = W2 * a2 + b2
% h_Wb = a3 = f(z3)z2 = W1 * data + repmat(b1, [, m]);
a2 = sigmoid(z2);
z3 = W2 * a2 + repmat(b2, [, m]);
a3 = z3;rhohats = mean(a2,);
rho = sparsityParam;
KLsum = sum(rho * log(rho ./ rhohats) + (-rho) * log((-rho) ./ (-rhohats)));squares = (a3 - data).^;
squared_err_J = (/) * (/m) * sum(squares(:)); %均方差项
weight_decay_J = (lambda/) * (sum(W1(:).^) + sum(W2(:).^));%权重衰减项
sparsity_J = beta * KLsum; %惩罚项cost = squared_err_J + weight_decay_J + sparsity_J;%损失函数值% delta3 = -(data - a3) .* fprime(z3);
% but fprime(z3) = a3 * (-a3)
delta3 = -(data - a3);
beta_term = beta * (- rho ./ rhohats + (-rho) ./ (-rhohats));
delta2 = ((W2' * delta3) + repmat(beta_term, [1,m]) ) .* a2 .* (1-a2);W2grad = (/m) * delta3 * a2' + lambda * W2; % W2梯度
b2grad = (/m) * sum(delta3, ); % b2梯度
W1grad = (/m) * delta2 * data' + lambda * W1; % W1梯度
b1grad = (/m) * sum(delta2, ); % b1梯度%-------------------------------------------------------------------
% Convert weights and bias gradients to a compressed form
% This step will concatenate and flatten all your gradients to a vector
% which can be used in the optimization method.
grad = [W1grad(:) ; W2grad(:) ; b1grad(:) ; b2grad(:)];end
%-------------------------------------------------------------------
% We are giving you the sigmoid function, you may find this function
% useful in your computation of the loss and the gradients.
function sigm = sigmoid(x) sigm = ./ ( + exp(-x));
end
displayColorNetwork.m
function displayColorNetwork(A)% display receptive field(s) or basis vector(s) for image patches
%
% A the basis, with patches as column vectors% In case the midpoint is not set at , we shift it dynamically
if min(A(:)) >=
A = A - mean(A(:)); % 0均值化
endcols = round(sqrt(size(A, )));% 每行大图像中小图像块的个数channel_size = size(A,) / ;
dim = sqrt(channel_size); % 小图像块内每行或列像素点个数
dimp = dim+;
rows = ceil(size(A,)/cols); % 每列大图像中小图像块的个数
B = A(:channel_size,:); % R通道像素值
C = A(channel_size+:channel_size*,:); % G通道像素值
D = A(*channel_size+:channel_size*,:); % B通道像素值
B=B./(ones(size(B,),)*max(abs(B)));% 归一化
C=C./(ones(size(C,),)*max(abs(C)));
D=D./(ones(size(D,),)*max(abs(D)));
% Initialization of the image
I = ones(dim*rows+rows-,dim*cols+cols-,);%Transfer features to this image matrix
for i=:rows-
for j=:cols- if i*cols+j+ > size(B, )
break
end % This sets the patch
I(i*dimp+:i*dimp+dim,j*dimp+:j*dimp+dim,) = ...
reshape(B(:,i*cols+j+),[dim dim]);
I(i*dimp+:i*dimp+dim,j*dimp+:j*dimp+dim,) = ...
reshape(C(:,i*cols+j+),[dim dim]);
I(i*dimp+:i*dimp+dim,j*dimp+:j*dimp+dim,) = ...
reshape(D(:,i*cols+j+),[dim dim]); end
endI = I + ; % 使I的范围从[-,]变为[，]
I = I / ; % 使I的范围从[，]变为[, ]
imagesc(I);
axis equal % 等比坐标轴：设置屏幕高宽比，使得每个坐标轴的具有均匀的刻度间隔
axis off % 关闭所有的坐标轴标签、刻度、背景end
参考资料
http://www.cnblogs.com/tornadomeet/archive/2013/04/08/3007435.html
http://www.cnblogs.com/tornadomeet/archive/2013/03/25/2980766.html
