1、Sparse Autoencoder
没想到，时隔多年后，再次搞这UFLDL，快速撸过吧。哎。
之前这第一个练习，没搞懂，现在感觉主要是这个推导并不完整，来的有点突然，要结合Tom的《机器学习》一起看，才能看懂。而且，其实这UFLDL的这一块BP就是看Tom的《机器学习》整出来的。主要是，之前一直没有搞懂这个残差这个概念，现在看Tom的《机器学习》算是看懂了。
，这个就是对于一个单元的所有输入整的一个概念。不过总感觉UFLDL的这块推导不好。当年可能也就是这一块卡住了，哎。
- 问题：
1）对于BP这块代码，其实自己编写还是有点困难，矩阵化运算完全编写，以后自己再重新编写过一遍。
2）由于对于MATLAB不熟，整体代码还是有很多不了解。
3）
4）
5）
本练习的代码为sparseae_exercise.zip，需要补充sampleIMAGES.m sparseAutoencoderCost.m computeNumericalGradient.m中的代码。
整体代码如下：
其中lxsparseae_exercise.m为自己练习，查看资源的代码。
其中IMAGES，有10幅512x512的灰白图像（已经被白化处理过）。保存为double类型，所以，整出负值来了。
其实以前感觉推导有问题，现在总结，主要是有两个原因。1、UFLDL对于BP的描述太简单，感觉完整推导不懂。2、Maltab不熟，对于里面有些语句，从面上都没办法完全理解。
虽然对于里面的细节语句还是不懂，但是总体现在就这样吧。
下面是这次实验的几个代码：sampleIMAGES.m
function patches = sampleIMAGES()
% sampleIMAGES
% Returns 10000 patches for training
load IMAGES; % load images from disk
patchsize = 8; % we'll use 8x8 patches
numpatches = 10000;
% Initialize patches with zeros. Your code will fill in this matrix--one
% column per patch, 10000 columns.
patches = zeros(patchsize*patchsize, numpatches);
%% ---------- YOUR CODE HERE --------------------------------------
% Instructions: Fill in the variable called "patches" using data
% from IMAGES.
%
% IMAGES is a 3D array containing 10 images
% For instance, IMAGES(:,:,6) is a 512x512 array containing the 6th image,
% and you can type "imagesc(IMAGES(:,:,6)), colormap gray;" to visualize
% it. (The contrast on these images look a bit off because they have
% been preprocessed using using "whitening." See the lecture notes for
% more details.) As a second example, IMAGES(21:30,21:30,1) is an image
% patch corresponding to the pixels in the block (21,21) to (30,30) of
% Image 1
%这图像已经经过白化处理，所以里面有整数和负数
% tic;
for imageNum=1:size(IMAGES,3)%遍历每一张图像
%虽然知道了图像尺寸，但希望得到一个通用的程序
[sizeRow,sizeCol]=size(IMAGES(:,:,imageNum));
for patchNum=1:numpatches/size(IMAGES,3)%这样整，有点犯贱。。。
%{
randx=randi(512-8+1);%取随机数，由于已经知道了图像的尺寸，所以就直接带入了数值，
randy=randi(512-8+1);
%1、
% tempPatch=IMAGES(randx:randx+7,randy:randy+7,imageNum);%取小块
% patches(:,(imageNum-1)*1000+patchNum)=reshape(tempPatch,64,1);%这reshape必须带上两个维度
%2、
% patches(:,(imageNum-1)*1000+patchNum)=reshape(IMAGES(randx:randx+7,randy:randy+7,imageNum),64,1);%这reshape必须带上两个维度
% 2相对于1，取消了赋予的中间变量tempPatch，运行速度更快一点。所以没必要的中间变量不要创建
%}
%3、
% patches(:,(imageNum-1)*1000+patchNum)=reshape(IMAGES(randi(512-8+1):randi(512-8+1)+7,randi(512-8+1):randi(512-8+1)+7,imageNum),64,1);%这reshape必须带上两个维度
% 上面这个代码不行，因为reshape的元素不能为可变值。违反语法规则，没办法执行。
%4、
%可以直接减8，但是想得到通用程序
xPos=randi(sizeRow-patchsize+1);
yPos=randi(sizeCol-patchsize+1);
patches(:,(imageNum-1)*1000+patchNum)=reshape(IMAGES(xPos:xPos+7,yPos:yPos+7,imageNum),64,1);
end
end
% toc;
% disp(['sampleIMAGES运行时间为：',num2str(toc)]);
%% ---------------------------------------------------------------
% For the autoencoder to work well we need to normalize the data
% Specifically, since the output of the network is bounded between [0,1]
% (due to the sigmoid activation function), we have to make sure
% the range of pixel values is also bounded between [0,1]
patches = normalizeData(patches);
end
%% ---------------------------------------------------------------
function patches = normalizeData(patches)
% Squash data to [0.1, 0.9] since we use sigmoid as the activation
% function in the output layer
% Remove DC (mean of images).
patches = bsxfun(@minus, patches, mean(patches));
% Truncate to +/-3 standard deviations and scale to -1 to 1
pstd = 3 * std(patches(:));
patches = max(min(patches, pstd), -pstd) / pstd;
% Rescale from [-1,1] to [0.1,0.9]
patches = (patches + 1) * 0.4 + 0.1;
end
sparseAutoencoderCost.m 这个函数，结合博客 SparseAutoEncoder 稀疏编码详解（Andrew ng课程系列）里面的描述，比较好理解。虽然对于矩阵方面的运算还是不熟，但是先就这样了。
STEP 2: 实现 sparseAutoencoderCost，稀疏损失函数
1）通过反转向量，把向量还原为权值矩阵
2）计算损失函数和梯度
由于画图和矩阵的原因，原稿使用ppt来制作的，现在就只能通过贴图的方式展现了，有需要ppt的可以私信我
sparseAutoencoderCost.m
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尺寸为[hiddenSize, visibleSize],W2尺寸为[visibleSize, hiddenSize]
%b1尺寸为[hiddenSize, 1],b2尺寸为[visibleSize, 1]
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.
%
%自己直接还是编不出来呀，还要参考被人的博客，哎。
%其实就这个过程维度的变换比较复杂，还理解不到位，看网页
%http://www.cnblogs.com/yymn/articles/4956333.html
%中对于维度变化的过程描述非常好。
J_cost=0;%直接的误差
J_weight=0;%权值衰减
J_sparse=0;%稀疏惩罚
%得到数据的维度和样本个数
[~,dataNum]=size(data);
%前向传播
%W1尺寸为[hiddenSize, visibleSize],data尺寸为[visibleSize, dataNum]
%b1尺寸为[hiddenSize,1]
%z2尺寸为[hiddenSize, dataNum]
z2=W1*data+repmat(b1,1,dataNum);
a2=sigmoid(z2);
%W2尺寸为[visibleSize, hiddenSize],b2尺寸为[visibleSize, 1]
%z3尺寸为[visibleSize, dataNum]
z3=W2*a2+repmat(b2,1,dataNum);
a3=sigmoid(z3);
%计算直接的误差
%sum对于矩阵是先进行列加，下面这个正好是每个样本，每个维度差值先相加，然后再累加
J_cost=(0.5/dataNum)*sum(sum((a3-data).^2));
%计算权值衰减项
%由于是平方的累加和，所以不用考虑那么多累加顺序
%W1尺寸为[hiddenSize, visibleSize]，W2尺寸为[visibleSize, hiddenSize]
%sum默认是先累加列，然后再加行，等于先累加前面层和后面层所有连接的系数到前面每个神经元
%这样其实和UFLDL中的公式是一致的
J_weight=(1/2)*(sum(sum(W1.^2))+sum(sum(W2.^2)));
%稀疏惩罚
%先要计算自编码网络自身的激活度，由于是一个三层的网络，所以就是第二层的激活度
%如果是n层的网络，就是2到n-2层的激活度
%就这还得看前面 自编码算法与稀疏性的章节 才能看懂
%由于a2尺寸为[hiddenSize, dataNum]，所以sum要是行累加，再除以样本个数，就是每个隐单元的激活度
%rho尺寸为[hiddenSize, 1]
%也有代码是 (1/m).*sum(a2,2); 但感觉下面这种格式更简单
rho=sum(a2,2)/dataNum;
%一开始也想着 .* 和 ./ 后面发现，一个常数和矩阵的乘除 直接 * /就可以了。
%一开始也是整个一堆的公司，但感觉这样太乱了，不方便查看，调用一个函数更好。
%J_sparse=sum(sparsityParam*log(sparsityParam/rho)+(1-sparsityParam)*log((1-sparsityParam)/(1-rho)));
J_sparse=KL(sparsityParam,rho);
%计算总的代价函数
cost=J_cost+lambda*J_weight+beta*J_sparse;
%下面开始整BP
%先计算最后一层的delta
%delta是总误差相对于每一个神经元的输入的导数，但是总误差对于输入的导数
%还要根据链式法则，对于激活函数求导，所以才会出现下面这两项
%现在还是感觉UFLDL公式的推导是不完整的，对于delta的定义也不严谨，还是得看书
%delta3尺寸为[visibleSize, dataNum]
delta3=-(data-a3).*sigmoidDer(z3);
%计算残差中添加的稀疏项
%其实UFLDL中并没有给出加了稀疏后的详细推导，自己在同文件编写了一个查看损耗中稀疏和delta中稀疏项的小代码
%在这个代码中，其余对于损耗函数的稀疏项的图像和UFLDL中的图像一直，但是在delta中的项不一致
%所以对于下面这项不懂，以后学点稀疏的时候，再来看。
deltaSparsity=beta*(-sparsityParam./rho+(1-sparsityParam)./(1-rho));
%W2尺寸为[visibleSize, hiddenSize],delta3尺寸为[visibleSize, dataNum]
%deltaSparsity尺寸为[hiddenSize, 1],z2尺寸为[hiddenSize, dataNum]
%delta2尺寸为[hiddenSize, dataNum]
delta2=(W2'*delta3+repmat(deltaSparsity,1,dataNum)).*sigmoidDer(z2);
%计算W1grad
%W1grad尺寸为[hiddenSize, visibleSize]
W1grad=W1grad+delta2*data';
W1grad = (1/dataNum)*W1grad+lambda*W1;
%计算b1grad
b1grad = b1grad+sum(delta2,2);
b1grad = (1/dataNum)*b1grad;%注意b的偏导是一个向量，所以这里应该把每一行的值累加起来
%计算W2grad
W2grad=W2grad+delta3*a2';
W2grad = (1/dataNum)*W2grad+lambda*W2;
%计算b2grad
b2grad = b2grad+sum(delta3,2);
b2grad = (1/dataNum)*b2grad;%注意b的偏导是一个向量，所以这里应该把每一行的值累加起来
%-------------------------------------------------------------------
% 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
%上面是源程序自带的，下面是自己编写的
function kl = KL(p,pj)
%p为设定的常数，pj为隐单元的平均激活度
kl=sum(p.*log(p./pj)+(1.-p).*log((1.-p)./(1.-pj)));
end
function sigmDer = sigmoidDer(x)
sigmDer = sigmoid(x).*(1-sigmoid(x));
end
computeNumericalGradient.m 这个函数由于一开始 没理解看明白J是一个函数，所以看不懂。
function numgrad = computeNumericalGradient(J, theta)
% numgrad = computeNumericalGradient(J, theta)
% theta: a vector of parameters
% J: a function that outputs a real-number. Calling y = J(theta) will return the
% function value at theta.
% Initialize numgrad with zeros
numgrad = zeros(size(theta));
%% ---------- YOUR CODE HERE --------------------------------------
% Instructions:
% Implement numerical gradient checking, and return the result in numgrad.
% (See Section 2.3 of the lecture notes.)
% You should write code so that numgrad(i) is (the numerical approximation to) the
% partial derivative of J with respect to the i-th input argument, evaluated at theta.
% I.e., numgrad(i) should be the (approximately) the partial derivative of J with
% respect to theta(i).
%
% Hint: You will probably want to compute the elements of numgrad one at a time.
%这段要运行代码的时候，看看数据的维度，才比较清楚
epsilon=1e-4;
%theta是个 3289（64x25x2+64+25） x 1的向量，所以下面是size(theta,1)
n=size(theta,1);
E=eye(n);
for i=1:n
delta=E(:,i)*epsilon;
numgrad(i)=(J(theta+delta)-J(theta-delta))/(epsilon*2.0);
end
%% ---------------------------------------------------------------
end
train.m 加了一点备注。
clear;clc;close all;
%% CS294A/CS294W Programming Assignment Starter Code
% Instructions
% ------------
%
% This file contains code that helps you get started on the
% programming assignment. You will need to complete the code in sampleIMAGES.m,
% sparseAutoencoderCost.m and computeNumericalGradient.m.
% For the purpose of completing the assignment, you do not need to
% change the code in this file.
%
%%======================================================================
%% STEP 0: Here we provide the relevant parameters values that will
% allow your sparse autoencoder to get good filters; you do not need to
% change the parameters below.
visibleSize = 8*8; % number of input units
hiddenSize = 25; % number of hidden units
sparsityParam = 0.01; % desired average activation of the hidden units.
% (This was denoted by the Greek alphabet rho, which looks like a lower-case "p",
% in the lecture notes).
% lambda = 0;
lambda = 0.0001; % weight decay parameter
% beta = 0;
beta = 3; % weight of sparsity penalty term
%%======================================================================
%% STEP 1: Implement sampleIMAGES
%
% After implementing sampleIMAGES, the display_network command should
% display a random sample of 200 patches from the dataset
patches = sampleIMAGES;
display_network(patches(:,randi(size(patches,2),200,1)),8);
set(gcf,'NumberTitle','off');
set(gcf,'Name','原始图像中截取的小块');
% Obtain random parameters theta
%theta输出为3289x1的向量，由于这是一个三层自编码的输出
%3289=64x25x2+64+25
theta = initializeParameters(hiddenSize, visibleSize);
%%======================================================================
%% STEP 2: Implement sparseAutoencoderCost
%
% You can implement all of the components (squared error cost, weight decay term,
% sparsity penalty) in the cost function at once, but it may be easier to do
% it step-by-step and run gradient checking (see STEP 3) after each step. We
% suggest implementing the sparseAutoencoderCost function using the following steps:
%
% (a) Implement forward propagation in your neural network, and implement the
% squared error term of the cost function. Implement backpropagation to
% compute the derivatives. Then (using lambda=beta=0), run Gradient Checking
% to verify that the calculations corresponding to the squared error cost
% term are correct.
%
% (b) Add in the weight decay term (in both the cost function and the derivative
% calculations), then re-run Gradient Checking to verify correctness.
%
% (c) Add in the sparsity penalty term, then re-run Gradient Checking to
% verify correctness.
%
% Feel free to change the training settings when debugging your
% code. (For example, reducing the training set size or
% number of hidden units may make your code run faster; and setting beta
% and/or lambda to zero may be helpful for debugging.) However, in your
% final submission of the visualized weights, please use parameters we
% gave in Step 0 above.
[costBegin, grad] = sparseAutoencoderCost(theta, visibleSize, hiddenSize, lambda, ...
sparsityParam, beta, patches);
%%======================================================================
%下面这个为检查函数computeNumericalGradient和函数sparseAutoencoderCost的代码
%检查完毕，就可以注释掉，不用再运行
%{
%% STEP 3: Gradient Checking
%
% Hint: If you are debugging your code, performing gradient checking on smaller models
% and smaller training sets (e.g., using only 10 training examples and 1-2 hidden
% units) may speed things up.
% First, lets make sure your numerical gradient computation is correct for a
% simple function. After you have implemented computeNumericalGradient.m,
% run the following:
%检查梯度的函数 computeNumericalGradient.m，开始编写computeNumericalGradient的时候开始验证
%然后就不用再检查
%checkNumericalGradient();
% Now we can use it to check your cost function and derivative calculations
% for the sparse autoencoder.
%下面这句语法上理解,其实一开始不懂，后面问了王鑫师姐，就完全懂了。
%下面的函数有两个参数，用逗号作为间隔，前一个参数是函数，后一个就是一个常数
%看 computeNumericalGradient(J, theta) 的函数定义，前一个参数J就是一个函数。
%这里就是用一个匿名函数作为参数，然后定义的x作为调用函数中的某一个参数变量
%所以后面如果调用这个参数的函数，那么送入的变量就会替换掉x
%搞懂这块真不容易呀，不过弄懂了是很爽呀，真就看懂了
%不过当年MATLAB也不熟，看这块代码确实是有点费劲
numgrad = computeNumericalGradient( @(x) sparseAutoencoderCost(x, visibleSize, ...
hiddenSize, lambda, ...
sparsityParam, beta, ...
patches), theta);
%下面就直接输出和theta[3289,1]维数一样的两个向量
% Use this to visually compare the gradients side by side
disp([numgrad grad]);
% Compare numerically computed gradients with the ones obtained from backpropagation
diff = norm(numgrad-grad)/norm(numgrad+grad);
fprintf('Norm of the difference between numerical and analytical gradient (should be < 1e-9)\n\n');
disp(diff); % Should be small. In our implementation, these values are
% usually less than 1e-9.
% When you got this working, Congratulations!!!
%%======================================================================
%}
%% STEP 4: After verifying that your implementation of
% sparseAutoencoderCost is correct, You can start training your sparse
% autoencoder with minFunc (L-BFGS).
% Randomly initialize the parameters
theta = initializeParameters(hiddenSize, visibleSize);
% Use minFunc to minimize the function
addpath minFunc/
options.Method = 'lbfgs'; % Here, we use L-BFGS to optimize our cost
% function. Generally, for minFunc to work, you
% need a function pointer with two outputs: the
% function value and the gradient. In our problem,
% sparseAutoencoderCost.m satisfies this.
options.maxIter = 400; % Maximum number of iterations of L-BFGS to run
options.display = 'on';
[opttheta, costEnd] = minFunc( @(p) sparseAutoencoderCost(p, ...
visibleSize, hiddenSize, ...
lambda, sparsityParam, ...
beta, patches), ...
theta, options);
%%======================================================================
%% STEP 5: Visualization
W1 = reshape(opttheta(1:hiddenSize*visibleSize), hiddenSize, visibleSize);
figure;
display_network(W1', 12);
set(gcf,'NumberTitle','off');
set(gcf,'Name','稀疏自编码后的第一层的权系数');
print -djpeg weights.jpg % save the visualization to a file
实验结果，就是这样类似于Gabor滤波器的图像。
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