UFLDL教程: Exercise:Learning color features with Sparse Autoencoders

Linear Decoders

Deep Learning and Unsupervised Feature Learning Tutorial Solutions

以三层的稀疏编码神经网络而言，在sparse autoencoder中的输出层满足下面的公式



从公式中可以看出，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。此时，也就是说输出层满足下面公式：



一个 S 型或 tanh 隐含层以及线性输出层构成的自编码器，我们称为线性解码器。

随着输出单元的激励函数的改变，这个输出单元梯度也相应变化。回顾之前每一个输出单元误差项定义为：



其中 y = x 是所期望的输出,

是自编码器的输出,

是激励函数.因为在输出层激励函数为 f(z) = z, 这样 f’(z) = 1，所以上述公式可以简化为



当然，若使用反向传播算法来计算隐含层的误差项时:



因为隐含层采用一个 S 型（或 tanh）的激励函数 f,在上述公式中，

依然是 S 型（或 tanh）函数的导数。

这样在用ＢＰ算法进行梯度的求解时，只需要更改误差的计算公式而已，改成如下公式





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。

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 0: Initialization
%  Here we initialize some parameters used for the exercise.

imageChannels = 3;     % number of channels (rgb, so 3)

patchDim   = 8;          % patch dimension
numPatches = 100000;   % number of patches

visibleSize = patchDim * patchDim * imageChannels;  % number of input units
outputSize  = visibleSize;   % number of output units
hiddenSize  = 400;           % number of hidden units

sparsityParam = 0.035; % desired average activation of the hidden units.
lambda = 3e-3;         % weight decay parameter
beta = 5;              % weight of sparsity penalty term

epsilon = 0.1;         % epsilon for ZCA whitening

%%======================================================================
%% STEP 1: 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 patches

debugHiddenSize = 5;
debugvisibleSize = 8;
patches = rand([8 10]);%随机产生10个样本，每个样本为一个8维的列向量，元素值为0~1
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-9.
disp(diff);

assert(diff < 1e-9, 'Difference too large. Check your gradient computation again');

% NOTE: Once your gradients check out, you should run step 0 again to
%       reinitialize the parameters
%}

%%======================================================================
%% STEP 2: Learn features on small patches从pathes中学习特征
%  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: Load patches 加载数据
%  In this step, we load 100k patches sampled from the STL10 dataset and
%  visualize them. Note that these patches have been scaled to [0,1]

load stlSampledPatches.mat  %里面自己定义了变量patches的值

displayColorNetwork(patches(:, 1:100));

%% STEP 2b: Apply preprocessing预处理
%  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, 2); %注意这里减掉的是每一维属性的均值
%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);%每一维都均值化

% Apply ZCA whitening
sigma = patches * patches' / numPatches;%协方差矩阵
[u, s, v] = svd(sigma);
ZCAWhite = u * diag(1 ./ sqrt(diag(s) + epsilon)) * u';%求出ZCAWhitening矩阵
patches = ZCAWhite * patches;

displayColorNetwork(patches(:, 1:100));

%% 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 45 minutes.

theta = initializeParameters(hiddenSize, visibleSize);

% Use minFunc to minimize the function
addpath minFunc/

options = struct;
options.Method = 'lbfgs';
options.maxIter = 400;
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 features

W = reshape(optTheta(1:visibleSize * hiddenSize), hiddenSize, visibleSize);
b = optTheta(2*hiddenSize*visibleSize+1:2*hiddenSize*visibleSize+hiddenSize);
figure;
%这里为什么要用(W*ZCAWhite)'呢？首先，使用W*ZCAWhite是因为每个样本x输入网络，
%其输出等价于W*ZCAWhite*x；另外，由于W*ZCAWhite的每一行才是一个隐含节点的变换值
%而displayColorNetwork函数是把每一列显示一个小图像块的，所以需要对其转置。

displayColorNetwork( (W*ZCAWhite)');

sparseAutoencoderLinearCost.m

function [cost,grad,features] = sparseAutoencoderLinearCost(theta, visibleSize, hiddenSize, ...
lambda, sparsityParam, beta, data)
% -------------------- 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 --------------------

%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

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);

% Loss and gradient variables (your code needs to compute these values)
m = size(data, 2); % 样本数量

%% ---------- YOUR CODE HERE --------------------------------------
%  Instructions: Compute the loss for the Sparse Autoencoder and gradients
%                W1grad, W2grad, b1grad, b2grad
%
%  Hint: 1) data(:,i) is the i-th example
%        2) 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, [1, m]);
a2 = sigmoid(z2);
z3 = W2 * a2 + repmat(b2, [1, m]);
a3 = z3;

rhohats = mean(a2,2);
rho = sparsityParam;
KLsum = sum(rho * log(rho ./ rhohats) + (1-rho) * log((1-rho) ./ (1-rhohats)));

squares = (a3 - data).^2;
squared_err_J = (1/2) * (1/m) * sum(squares(:));              %均方差项
weight_decay_J = (lambda/2) * (sum(W1(:).^2) + sum(W2(:).^2));%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 * (1-a3)
delta3 = -(data - a3);
beta_term = beta * (- rho ./ rhohats + (1-rho) ./ (1-rhohats));
delta2 = ((W2' * delta3) + repmat(beta_term, [1,m]) ) .* a2 .* (1-a2);

W2grad = (1/m) * delta3 * a2' + lambda * W2;   % W2梯度
b2grad = (1/m) * sum(delta3, 2);               % b2梯度
W1grad = (1/m) * delta2 * data' + lambda * W1; % W1梯度
b1grad = (1/m) * sum(delta2, 2);               % 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(:)];

%-------------------------------------------------------------------
% 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 = 1 ./ (1 + exp(-x));
end

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 0, we shift it dynamically
if min(A(:)) >= 0
A = A - mean(A(:));%0均值化
end

cols = round(sqrt(size(A, 2)));% 每行大图像中小图像块的个数

channel_size = size(A,1) / 3;
dim = sqrt(channel_size);% 小图像块内每行或列像素点个数
dimp = dim+1;
rows = ceil(size(A,2)/cols);% 每列大图像中小图像块的个数
B = A(1:channel_size,:);% R通道像素值
C = A(channel_size+1:channel_size*2,:);% G通道像素值
D = A(2*channel_size+1:channel_size*3,:);% B通道像素值
B=B./(ones(size(B,1),1)*max(abs(B)));% 归一化
C=C./(ones(size(C,1),1)*max(abs(C)));
D=D./(ones(size(D,1),1)*max(abs(D)));
% Initialization of the image
I = ones(dim*rows+rows-1,dim*cols+cols-1,3);

%Transfer features to this image matrix
for i=0:rows-1
for j=0:cols-1

if i*cols+j+1 > size(B, 2)
break
end

% This sets the patch
I(i*dimp+1:i*dimp+dim,j*dimp+1:j*dimp+dim,1) = ...
reshape(B(:,i*cols+j+1),[dim dim]);
I(i*dimp+1:i*dimp+dim,j*dimp+1:j*dimp+dim,2) = ...
reshape(C(:,i*cols+j+1),[dim dim]);
I(i*dimp+1:i*dimp+dim,j*dimp+1:j*dimp+dim,3) = ...
reshape(D(:,i*cols+j+1),[dim dim]);

end
end

I = I + 1;% 使I的范围从[-1,1]变为[0，2]
I = I / 2;% 使I的范围从[0，2]变为[0, 1]
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

参考文献

Exercise:Learning color features with Sparse Autoencoders

Deep learning：十七(Linear Decoders，Convolution和Pooling)

线性解码器

吴恩达 Andrew Ng 的公开课