UFIDL稀疏自编码代码实现及解释
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UFIDL稀疏自编码代码实现及解释
1.今天我们来讲一下UFIDL的第一个练习。
1.我们来看看最难的一个.m文件
%% ---------- 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. % % size(data, 1) % 64% size(W1) % 25 64% size(W2) % 64 25% size(b1) % 25 1% size(b2) % 64 1%样本的个数m = size(data, 2);%前向传播%第二层的输入值z_2 = W1 * data + repmat(b1, 1, m);%第二层的激活值a_2 = sigmoid(z_2); % 25 10000%计算rho_hat的值,其值为第二层的每个维度(64)上的平均激活度,sum(a_2,2)表示对第二维求和,即对每一行求和rho_hat = sum(a_2, 2) / m; % This doesn't contain an x because the data % above "has" the x%第三层的输入z_3 = W2 * a_2 + repmat(b2, 1, m);%第三层的激活a_3 = sigmoid(z_3); % 64 10000%激活值与实际值的差diff = a_3 - data;%稀疏自编码的惩罚,KLdivergence(相对熵)sparse_penalty = kl(sparsityParam, rho_hat);%简化的代价函数%这个写的比较简洁J_simple = sum(sum(diff.^2)) / (2*m);%正则项,即所有W元素的平方和%这个写得也不错,W1(:)使得矩阵的元素按列重新排布reg = sum(W1(:).^2) + sum(W2(:).^2);%总的代价cost = J_simple + beta * sparse_penalty + lambda * reg / 2;% Backpropogationdelta_3 = diff .* (a_3 .* (1-a_3)); % 64 10000%计算残差2的基本部分d2_simple = W2' * delta_3; % 25 10000%计算残差2的惩罚d2_pen = kl_delta(sparsityParam, rho_hat);%计算残差2delta_2 = (d2_simple + beta * repmat(d2_pen,1, m)) .* a_2 .* (1-a_2);%计算b的梯度b2grad = sum(delta_3, 2)/m;b1grad = sum(delta_2, 2)/m;%计算W的梯度W2grad = delta_3 * a_2'/m + lambda * W2; % 25 64W1grad = delta_2 * data'/m + lambda * W1; % 25 64%-------------------------------------------------------------------% 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%相对熵function ans = kl(r, rh) ans = sum(r .* log(r ./ rh) + (1-r) .* log( (1-r) ./ (1-rh)));end%相对熵的残差计算公式function ans = kl_delta(r, rh) ans = -(r./rh) + (1-r) ./ (1-rh);end%sigmoid函数的导数function pr = prime(x) pr = sigmoid(x) .* (1 - sigmoid(x));end
我对上述进行了解释。
接下去是SampleIMAGES文件,这个比较简单我就不解释了
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 1% imagesc(IMAGES(:,:,10)) % 1-10% size(patches) % 64 x 10000% size(patches(:,1)) % 64 x 1for i = 1:numpatches x = randi(512-8+1); y = randi(512-8+1); sample = IMAGES(x:x+7,y:y+7,randi(10)); patches(:,i) = sample(:);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从10幅图像里面随机抽取10000个图像块,并进行了一些归一化处理。
接下去是计算梯度函数
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 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. % size(theta) 2 1 | 3289 1% size(numgrad) 2 1 | 3289 1eps = 1e-4;n = size(numgrad);I = eye(n, n);for i = 1:size(numgrad) %通过单位矩阵来构造向量,比较有趣 eps_vec = I(:,i) * eps; numgrad(i) = (J(theta + eps_vec) - J(theta - eps_vec)) / (2 * eps);end% numgrad = (J(theta + eps) - J(theta - eps)) / (2 * eps)%% ---------------------------------------------------------------end最后我们运行官方的train文件,并等待结果(大约1分钟左右)
我们可以看到matlab的输出
可以看到我们我们通过BFGS(共轭梯度下降法),总共迭代了400次,花了51秒。
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