Exercise:Sparse Autoencoder
习题的链接：
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：
我的实现：
sampleIMAGES.m
function patches = sampleIMAGES()% sampleIMAGES% Returns 10000 patches for trainingload 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 1for 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 1pstd = 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 zerosnumgrad = 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 matrixW1 = reshape(theta(1:hiddenSize*visibleSize), hiddenSize, visibleSize);% W2 is a visibleSize * hiddenSize matrixW2 = reshape(theta(hiddenSize*visibleSize+1:2*hiddenSize*visibleSize), visibleSize, hiddenSize);% b1 is a hiddenSize * 1 vectorb1 = theta(2*hiddenSize*visibleSize+1:2*hiddenSize*visibleSize+hiddenSize);% b2 is a visible * 1 vectorb2 = 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 propagationz2 = W1 * data + repmat(b1, 1, numCases);a2 = sigmoid(z2);z3 = W2 * a2 + repmat(b2, 1, numCases);a3 = sigmoid(z3);% errorsqrerror = (data - a3) .* (data - a3);error = sum(sum(sqrerror)) / (2 * numCases);% weight decaywtdecay = (sum(sum(W1 .* W1)) + sum(sum(W2 .* W2))) / 2;% sparsityrho = 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 matrixdelta3 = -(data - a3) .* sigmoiddiff(z3);% delta2 is a hiddenSize * numCases matrixsparsityterm = 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));endfunction sigmdiff = sigmoiddiff(x) sigmdiff = sigmoid(x) .* (1 - sigmoid(x));end
最终训练结果：
