Exercise:Sparse Autoencoder
习题的链接：Exercise:Sparse Autoencoder
注意点：
1、训练样本像素值需要归一化。
因为输出层的激活函数是logistic函数，值域(0,1)，
如果训练样本每个像素点没有进行归一化，那将无法进行自编码。
2、训练阶段，向量化实现比for循环实现快十倍。
3、最后产生的图片阵列是将W1权值矩阵的转置，每一列作为一张图片。
第i列其实就是最大可能激活第i个隐藏节点的图片xi，再乘以常数因子C(其中C就是W1第i行元素的平方和)。
证明可见：Visualizing a Trained Autoencoder
我的实现：
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 for i=1:numpatches % generate random row&col number [1, 512-patchsize+1=505] % generate random IMAGES id [1, 10] row = round(1 + rand(1,1)*504); col = round(1 + rand(1,1)*504); pid = round(1 + rand(1,1)*9); patches(:, i) = reshape(IMAGES(row:row+7, col:col+7, pid), patchsize*patchsize, 1); end %% --------------------------------------------------------------- % 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
computeNumericalGradient.m
function numgrad = computeNumericalGradient(J, theta) % numgrad = computeNumericalGradient(J, theta) % theta: a vector of parameters (column vector) % 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. N = size(theta, 1); EPSILON = 1e-4; Identity = eye(N); for i = 1:N numgrad(i,:) = (J(theta + EPSILON * Identity(:, i)) - J(theta - EPSILON * Identity(:, i))) / (2 * EPSILON); end %% --------------------------------------------------------------- end
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 is a hiddenSize * visibleSize matrix W1 = reshape(theta(1:hiddenSize*visibleSize), hiddenSize, visibleSize); % W2 is a visibleSize * hiddenSize matrix W2 = reshape(theta(hiddenSize*visibleSize+1:2*hiddenSize*visibleSize), visibleSize, hiddenSize); % b1 is a hiddenSize * 1 vector b1 = theta(2*hiddenSize*visibleSize+1:2*hiddenSize*visibleSize+hiddenSize); % b2 is a visible * 1 vector 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. % numCases = size(data, 2); % forward propagation z2 = W1 * data + repmat(b1, 1, numCases); a2 = sigmoid(z2); z3 = W2 * a2 + repmat(b2, 1, numCases); a3 = sigmoid(z3); % error sqrerror = (data - a3) .* (data - a3); error = sum(sum(sqrerror)) / (2 * numCases); % weight decay wtdecay = (sum(sum(W1 .* W1)) + sum(sum(W2 .* W2))) / 2; % sparsity rho = sum(a2, 2) ./ numCases; divergence = sparsityParam .* log(sparsityParam ./ rho) + (1 - sparsityParam) .* log((1 - sparsityParam) ./ (1 - rho)); sparsity = sum(divergence); cost = error + lambda * wtdecay + beta * sparsity; % delta3 is a visibleSize * numCases matrix delta3 = -(data - a3) .* sigmoiddiff(z3); % delta2 is a hiddenSize * numCases matrix sparsityterm = beta * (-sparsityParam ./ rho + (1-sparsityParam) ./ (1-rho)); delta2 = (W2' * delta3 + repmat(sparsityterm, 1, numCases)) .* sigmoiddiff(z2); W1grad = delta2 * data' ./ numCases + lambda * W1; b1grad = sum(delta2, 2) ./ numCases; W2grad = delta3 * a2' ./ numCases + lambda * W2; b2grad = sum(delta3, 2) ./ numCases; %------------------------------------------------------------------- % 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 function sigmdiff = sigmoiddiff(x) sigmdiff = sigmoid(x) .* (1 - sigmoid(x)); end
最终训练结果：
