sparse autoencoder的一个实例练习，这个例子所要实现的内容大概如下：从给定的很多张自然图片中截取出大小为8*8的小patches图片共10000张，现在需要用sparse autoencoder的方法训练出一个隐含层网络所学习到的特征。该网络共有3层，输入层是64个节点，隐含层是25个节点，输出层当然也是64个节点了。
main函数, 分五步走，每个函数的实现细节在下边都列出了。
1 %%====================================================================== 2 %% STEP 0: Here we provide the relevant parameters values that will 3 % allow your sparse autoencoder to get good filters; you do not need to 4 % change the parameters below. 5 6 visibleSize = 8*8; % number of input units 7 hiddenSize = 25; % number of hidden units 8 sparsityParam = 0.01; % desired average activation of the hidden units. 9 % (This was denoted by the Greek alphabet rho, 10 % which looks like a lower-case "p", 11 % in the lecture notes). 12 lambda = 0.0001; % weight decay parameter 13 beta = 3; % weight of sparsity penalty term 14 15 %%====================================================================== 16 %% STEP 1: Implement sampleIMAGES 17 % 18 % After implementing sampleIMAGES, the display_network command should 19 % display a random sample of 200 patches from the dataset 20 patches = sampleIMAGES; 21 display_network(patches(:,randi(size(patches,2),200,1)),8); 22 23 24 % Obtain random parameters theta 25 theta = initializeParameters(hiddenSize, visibleSize); 26 27 %%====================================================================== 28 %% STEP 2: Implement sparseAutoencoderCost 29 % 30 % You can implement all of the components (squared error cost, weight decay term, 31 % sparsity penalty) in the cost function at once, but it may be easier to do 32 % it step-by-step and run gradient checking (see STEP 3) after each step. We 33 % suggest implementing the sparseAutoencoderCost function using the following steps: 34 % 35 % (a) Implement forward propagation in your neural network, and implement the 36 % squared error term of the cost function. Implement backpropagation to 37 % compute the derivatives. Then (using lambda=beta=0), run Gradient Checking 38 % to verify that the calculations corresponding to the squared error cost 39 % term are correct. 40 % 41 % (b) Add in the weight decay term (in both the cost function and the derivative 42 % calculations), then re-run Gradient Checking to verify correctness. 43 % 44 % (c) Add in the sparsity penalty term, then re-run Gradient Checking to 45 % verify correctness. 46 % 47 % Feel free to change the training settings when debugging your 48 % code. (For example, reducing the training set size or 49 % number of hidden units may make your code run faster; and setting beta 50 % and/or lambda to zero may be helpful for debugging.) However, in your 51 % final submission of the visualized weights, please use parameters we 52 % gave in Step 0 above. 53 54 [cost, grad] = sparseAutoencoderCost(theta, visibleSize, hiddenSize, ... 55 lambda,sparsityParam, beta, patches); 56 57 %%====================================================================== 58 %% STEP 3: Gradient Checking 59 % 60 % Hint: If you are debugging your code, performing gradient checking on smaller models 61 % and smaller training sets (e.g., using only 10 training examples and 1-2 hidden 62 % units) may speed things up. 63 64 % First, lets make sure your numerical gradient computation is correct for a 65 % simple function. After you have implemented computeNumericalGradient.m, 66 % run the following: 67 checkNumericalGradient(); 68 69 % Now we can use it to check your cost function and derivative calculations 70 % for the sparse autoencoder. 71 numgrad = computeNumericalGradient( @(x) sparseAutoencoderCost(x, visibleSize, ... 72 hiddenSize, lambda,sparsityParam, beta, patches), theta); 73 74 % Use this to visually compare the gradients side by side 75 disp([numgrad grad]); 76 77 % Compare numerically computed gradients with the ones obtained from backpropagation 78 diff = norm(numgrad-grad)/norm(numgrad+grad); 79 disp(diff); % Should be small. In our implementation, these values are 80 % usually less than 1e-9. 81 % When you got this working, Congratulations!!! 82 83 %%====================================================================== 84 %% STEP 4: After verifying that your implementation of 85 % sparseAutoencoderCost is correct, You can start training your sparse 86 % autoencoder with minFunc (L-BFGS). 87 88 % Randomly initialize the parameters 89 theta = initializeParameters(hiddenSize, visibleSize); 90 91 % Use minFunc to minimize the function 92 addpath minFunc/ 93 options.Method = 'lbfgs'; % Here, we use L-BFGS to optimize our cost 94 % function. Generally, for minFunc to work, you 95 % need a function pointer with two outputs: the 96 % function value and the gradient. In our problem, 97 % sparseAutoencoderCost.m satisfies this. 98 options.maxIter = 400; % Maximum number of iterations of L-BFGS to run 99 options.display = 'on'; 100 [opttheta, cost] = minFunc( @(p) sparseAutoencoderCost(p,visibleSize, hiddenSize, ... 101 lambda, sparsityParam, beta, patches),theta, options); 102 %%====================================================================== 103 %% STEP 5: Visualization 104 105 W1 = reshape(opttheta(1:hiddenSize*visibleSize), hiddenSize, visibleSize); 106 display_network(W1', 12); 107 108 print -djpeg weights.jpg % save the visualization to a file 109 110 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 对应step1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%% 111 %三个函数（sampleIMAGES）（normalizeData）（initializeParameters）%%%% 112 function patches = sampleIMAGES() 113 load IMAGES; % 加载初始的10张512*512大图片 114 115 patchsize = 8; % 采样大小 116 numpatches = 10000; 117 118 % 初始化该矩阵为0，该矩阵为 64*10000维每一列为一张图片. 119 patches = zeros(patchsize*patchsize, numpatches); 120 121 % IMAGES 为一个包含10 张images的三维数组，IMAGES(:,:,6) 是一个第六张图片的 512x512 的二维数组, 122 % 命令 "imagesc(IMAGES(:,:,6)), colormap gray;" 可以把第六张图可视化. 123 % 这几张图是经过whiteing预处理的？ 124 % IMAGES(21:30,21:30,1) 就是从第一张图采样得到的(21,21) to (30,30) 的小patchs 125 126 %在每张图片中随机选取1000个patch，共10000个patch 127 for imageNum = 1:10 128 [rowNum colNum] = size(IMAGES(:,:,imageNum)); 129 %实现每张图片选取1000个patch 130 for patchNum = 1:1000 131 %得到左上角的两个点 132 xPos = randi([1,rowNum-patchsize+1]); 133 yPos = randi([1, colNum-patchsize+1]); 134 %填充到矩阵里 135 patches(:,(imageNum-1)*1000+patchNum) = ... 136 reshape(IMAGES(xPos:xPos+7,yPos:yPos+7,imageNum),64,1); 137 end 138 end 139 %由于autoencoder的激励函数是sigmod函数，输出值限定在[0,1],故为了达到H W,b（x）= x，x作为输入， 140 %也要限定在0-1之间，故需要进行正则化 141 patches = normalizeData(patches); 142 end 143 144 % 正则化的函数，不太明白s-sigma法则？ 145 function patches = normalizeData(patches) 146 % 减去均值 147 patches = bsxfun(@minus, patches, mean(patches)); 148 % s = std(X)，此处X是一个矢量，该函数返回标准偏差（注意其分母为n-1，而不是n） 。 149 % 结果s是一个X各样本偏差无偏估计的平方根(X包含独立的、同分布样本)。 150 % 如果X是一个矩阵，该函数返回一个行矢量，它包含了X每列元素的标准偏差。 151 pstd = 3 * std(patches(:)); 152 patches = max(min(patches, pstd), -pstd) / pstd; 153 % 重新压缩 从[-1,1] 到 [0.1,0.9] 154 patches = (patches + 1) * 0.4 + 0.1; 155 end 156 157 %首先初始化参数 158 function theta = initializeParameters(hiddenSize, visibleSize) 159 % Initialize parameters randomly based on layer sizes. 160 % we'll choose weights uniformly from the interval [-r, r] 161 r = sqrt(6) / sqrt(hiddenSize+visibleSize+1); 162 %rand(a,b)产生均匀分布的随机矩阵维度为a*b，元素取值范围0.0 ～1.0。 163 W1 = rand(hiddenSize, visibleSize) * 2 * r - r; 164 %rand(a,b)*2*r即取值范围为（0-2r）， rand(a,b)*2*r -r即取值范围为（-r - r） 165 W2 = rand(visibleSize, hiddenSize) * 2 * r - r; 166 b1 = zeros(hiddenSize, 1); %连接到hidden unit的偏置单元 167 b2 = zeros(visibleSize, 1); %链接到output layer的偏置单元 168 % 将矩阵合并为一个向量 169 theta = [W1(:) ; W2(:) ; b1(:) ; b2(:)]; 170 %初始化参数结束 171 end 172 173 174 175 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 对应step 2 %%%%%%%%%%%%%%%%%%%%%%%%%%%% 176 %%%%%返回稀疏损失函数的值与梯度值%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 177 function [cost,grad] = sparseAutoencoderCost(theta, visibleSize, hiddenSize, ... 178 lambda, sparsityParam, beta, data) 179 % visibleSize: 输入层单元数 180 % hiddenSize: 隐藏单元数 181 % lambda: 正则项 182 % sparsityParam: （p）指定的平均激活度p 183 % beta: 稀疏权重项B 184 % data: 64x10000 的矩阵为training data,data(:,i) 是第i个训练样例. 185 % 把参数拼接为一个向量，因为采用L-BFGS优化，L-BFGS要求的就是向量. 186 % 将长向量转换成每一层的权值矩阵和偏置向量值 187 % theta向量的的 1->hiddenSize*visibleSize，W1共hiddenSize*visibleSize 个元素，重新作为矩阵 188 W1 = reshape(theta(1:hiddenSize*visibleSize), hiddenSize, visibleSize); 189 190 %类似以上一直往后放 191 W2 = reshape(theta(hiddenSize*visibleSize+1:2*hiddenSize*visibleSize), visibleSize, hiddenSize); 192 b1 = theta(2*hiddenSize*visibleSize+1:2*hiddenSize*visibleSize+hiddenSize); 193 b2 = theta(2*hiddenSize*visibleSize+hiddenSize+1:end); 194 195 % 参数对应的梯度矩阵 ; 196 cost = 0; 197 W1grad = zeros(size(W1)); 198 W2grad = zeros(size(W2)); 199 b1grad = zeros(size(b1)); 200 b2grad = zeros(size(b2)); 201 202 Jcost = 0; %直接误差 203 Jweight = 0;%权值惩罚 204 Jsparse = 0;%稀疏性惩罚 205 [n m] = size(data); %m为样本的个数，n为样本的特征数 206 207 %前向算法计算各神经网络节点的线性组合值和active值 208 %W1为 hiddenSize*visibleSize的矩阵 209 %data为 visibleSize* trainexampleNum的矩阵 210 %remat(b1,1,m)把向量b1复制扩展为hiddenSize*m列 211 % 根据公式 Z^(l) = z^(l-1)*W^(l-1)+b^(l-1) 212 %z2保存的是10000个样本下隐藏层的输入，为hiddenSize*m维的矩阵，每一列代表一次输入 213 z2= W1*data + remat(b1,1,m)；%第二层的输入 214 a2 = sigmoid(z2); %对z2取sigmod 即得到a2，即隐藏层的输出 215 z3 = W2*a2+repmat(b2,1,m); %output layer 的输入 216 a3 = sigmoid(z3); %output 层的输出 217 218 % 计算预测产生的误差 219 %对应J(W,b), 外边的sum是对所有样本求和，里边的sum是对输出层的所有分量求和 220 Jcost = (0.5/m)*sum(sum((a3-data).^2)); 221 %计算权值惩罚项 正则化项，并没有带正则项参数 222 Jweight = (1/2)*(sum(sum(W1.^2))+sum(sum(W2.^2))); 223 %计算稀疏性规则项 sum(matrix,2)是进行按行求和运算，即所有样本在隐层的输出累加求均值 224 % rho为一个hiddenSize*1 维的向量 225 226 rho = (1/m).*sum(a2,2);%求出隐含层输出aj的平均值向量 rho为hiddenSize维的 227 %求稀疏项的损失 228 Jsparse = sum(sparsityParam.*log(sparsityParam./rho)+(1-sparsityParam).*log((1-sparsityParam)./(1-rho))); 229 %损失函数的总表达式 损失项 + 正则化项 + 稀疏项 230 cost = Jcost + lambda*Jweight + beta*Jsparse; 231 %计算l = 3 即 output-layer层的误差dleta3，因为在autoencoder中输入等于输出h(W,b)=x 232 delta3 = -(data-a3).*sigmoidInv(z3); 233 %因为加入了稀疏规则项，所以计算偏导时需要引入该项，sterm为稀疏项，为hiddenSize维的向量 234 sterm = beta*(-sparsityParam./rho+(1-sparsityParam)./(1-rho)) 235 % W2 为64*25的矩阵，d3为第三层的输出为64*10000的矩阵，每一列为每个样本x^(i)的输出，W2'为W2的转置 236 % repmat(sterm,1,m)会把函数复制扩展为m列的矩阵，每一列都为sterm向量。 237 % d2为hiddenSize*10000的矩阵 238 delta2 = (W2'*delta3+repmat(sterm,1,m)).*sigmoidInv(z2); 239 240 %计算W1grad 241 % data'为10000*64的矩阵 d2*data' 位25*64的矩阵 242 W1grad = W1grad+delta2*data'; 243 W1grad = (1/m)*W1grad+lambda*W1; 244 245 %计算W2grad 246 W2grad = W2grad+delta3*a2'; 247 W2grad = (1/m).*W2grad+lambda*W2; 248 249 %计算b1grad 250 b1grad = b1grad+sum(delta2,2); 251 b1grad = (1/m)*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 252 253 %计算b2grad 254 b2grad = b2grad+sum(delta3,2); 255 b2grad = (1/m)*b2grad; 256 %计算完成重新转为向量 257 grad = [W1grad(:) ; W2grad(:) ; b1grad(:) ; b2grad(:)]; 258 end 259 260 %------------------------------------------------------------------- 261 % Here's an implementation of the sigmoid function, which you may find useful 262 % in your computation of the costs and the gradients. This inputs a (row or 263 % column) vector (say (z1, z2, z3)) and returns (f(z1), f(z2), f(z3)). 264 265 function sigm = sigmoid(x) 266 sigm = 1 ./ (1 + exp(-x)); 267 end 268 269 %sigmoid函数的导函数 270 function sigmInv = sigmoidInv(x) 271 sigmInv = sigmoid(x).*(1-sigmoid(x)); 272 end 273 274 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 对应step 3 %%%%%%%%%%%%%%%%%%%%%%%%%%%% 275 %三个函数：（checkNumericalGradient）（simpleQuadraticFunction）（computeNumericalGradient） 276 function [] = checkNumericalGradient() 277 x = [4; 10]; 278 %当前简单函数实际的值与实际的导函数 279 [value, grad] = simpleQuadraticFunction(x); 280 % 在点 x 处计算简单函数的梯度，("@simpleQuadraticFunction" denotes a pointer to a function.) 281 numgrad = computeNumericalGradient(@simpleQuadraticFunction, x); 282 % disp()等价于 print() 283 disp([numgrad grad]); 284 fprintf('The above two columns you get should be very similar.\n(Left-Your Numerical Gradient, Right-Analytical Gradient)\n\n'); 285 % norm 等价于 sqrt(sum(X.^2)); 如果实现正确，设置 EPSILON = 0.0001，误差应该为2.1452e-12 286 diff = norm(numgrad-grad)/norm(numgrad+grad); 287 disp(diff); 288 fprintf('Norm of the difference between numerical and analytical gradient (should be < 1e-9)\n\n'); 289 end 290 291 %这个简单函数用来检验写的computeNumericalGradient函数的正确性 292 function [value,grad] = simpleQuadraticFunction(x) 293 % this function accepts a 2D vector as input. 294 % Its outputs are: 295 % value: h(x1, x2) = x1^2 + 3*x1*x2 296 % grad: A 2x1 vector that gives the partial derivatives of h with respect to x1 and x2 297 % Note that when we pass @simpleQuadraticFunction(x) to computeNumericalGradients, we're assuming 298 % that computeNumericalGradients will use only the first returned value of this function. 299 value = x(1)^2 + 3*x(1)*x(2); 300 grad = zeros(2, 1); 301 grad(1) = 2*x(1) + 3*x(2); 302 grad(2) = 3*x(1); 303 end 304 305 %梯度检验的函数 306 function numgrad = computeNumericalGradient(J, theta) 307 % theta: 参数，向量或者实数均可 308 % J: 输出值为实数的函数. 调用y = J(theta)将会返回函数在theta处的值 309 310 % numgrad初始化为0,与theta维度相同 311 numgrad = zeros(size(theta)); 312 EPSILON = 1e-4; 313 % theta是一个行向量，size(theta,1)是求行数 314 n = size(theta,1); 315 %产生一个维度为n的单位矩阵 316 E = eye(n); 317 for i = 1:n 318 % (n,:)代表第n行，所有的列 319 % (:,n)代表所有行，第n列 320 % 由于E是单位矩阵，所以只有第i行第i列的元素变为EPSILON 321 delta = E(:,i)*EPSILON; 322 %向量第i维度的值 323 numgrad(i) = (J(theta+delta)-J(theta-delta))/(EPSILON*2.0); 324 end 325 %% --------------------------------------------------------------- 326 327 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 对应step 5 %%%%%%%%%%%%%%%%%%%%%%%%%%%% 328 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%关于函数的展示%%%%%%%%%%%%%%%%%%%%%%%%%%% 329 function [h, array] = display_network(A, opt_normalize, opt_graycolor, cols, opt_colmajor) 330 % This function visualizes filters in matrix A. Each column of A is a 331 % filter. We will reshape each column into a square image and visualizes 332 % on each cell of the visualization panel. 333 % All other parameters are optional, usually you do not need to worry 334 % about it. 335 % opt_normalize: whether we need to normalize the filter so that all of 336 % them can have similar contrast. Default value is true. 337 % opt_graycolor: whether we use gray as the heat map. Default is true. 338 % cols: how many columns are there in the display. Default value is the 339 % squareroot of the number of columns in A. 340 % opt_colmajor: you can switch convention to row major for A. In that 341 % case, each row of A is a filter. Default value is false. 342 warning off all 343 344 if ~exist('opt_normalize', 'var') || isempty(opt_normalize) 345 opt_normalize= true; 346 end 347 348 if ~exist('opt_graycolor', 'var') || isempty(opt_graycolor) 349 opt_graycolor= true; 350 end 351 352 if ~exist('opt_colmajor', 'var') || isempty(opt_colmajor) 353 opt_colmajor = false; 354 end 355 356 % rescale 357 A = A - mean(A(:)); 358 359 if opt_graycolor, colormap(gray); end 360 361 % compute rows, cols 362 [L M]=size(A); 363 sz=sqrt(L); 364 buf=1; 365 if ~exist('cols', 'var') 366 if floor(sqrt(M))^2 ~= M 367 n=ceil(sqrt(M)); 368 while mod(M, n)~=0 && n<1.2*sqrt(M), n=n+1; end 369 m=ceil(M/n); 370 else 371 n=sqrt(M); 372 m=n; 373 end 374 else 375 n = cols; 376 m = ceil(M/n); 377 end 378 379 array=-ones(buf+m*(sz+buf),buf+n*(sz+buf)); 380 381 if ~opt_graycolor 382 array = 0.1.* array; 383 end 384 385 386 if ~opt_colmajor 387 k=1; 388 for i=1:m 389 for j=1:n 390 if k>M, 391 continue; 392 end 393 clim=max(abs(A(:,k))); 394 if opt_normalize 395 array(buf+(i-1)*(sz+buf)+(1:sz),buf+(j-1)*(sz+buf)+(1:sz))=reshape(A(:,k),sz,sz)/clim; 396 else 397 array(buf+(i-1)*(sz+buf)+(1:sz),buf+(j-1)*(sz+buf)+(1:sz))=reshape(A(:,k),sz,sz)/max(abs(A(:))); 398 end 399 k=k+1; 400 end 401 end 402 else 403 k=1; 404 for j=1:n 405 for i=1:m 406 if k>M, 407 continue; 408 end 409 clim=max(abs(A(:,k))); 410 if opt_normalize 411 array(buf+(i-1)*(sz+buf)+(1:sz),buf+(j-1)*(sz+buf)+(1:sz))=reshape(A(:,k),sz,sz)/clim; 412 else 413 array(buf+(i-1)*(sz+buf)+(1:sz),buf+(j-1)*(sz+buf)+(1:sz))=reshape(A(:,k),sz,sz); 414 end 415 k=k+1; 416 end 417 end 418 end 419 420 if opt_graycolor 421 h=imagesc(array,'EraseMode','none',[-1 1]); 422 else 423 h=imagesc(array,'EraseMode','none',[-1 1]); 424 end 425 axis image off 426 427 drawnow; 428 429 warning on all