前言：
本節是練習Linear decoder的應用，關於Linear decoder的相關知識介紹請參考：Deep learning：十七(Linear Decoders，Convolution和Pooling)，實驗步驟參考Exercise: Implement deep networks for digit classification。本次實驗是用linear decoder的sparse autoencoder來訓練出stl-10數據庫圖片的patch特征。並且這次的訓練權值是針對rgb圖像塊的。
基礎知識：
PCA Whitening是保證數據各維度的方差為1，而ZCA Whitening是保證數據各維度的方差相等即可，不一定要唯一。並且這兩種whitening的一般用途也不一樣，PCA Whitening主要用於降維且去相關性，而ZCA Whitening主要用於去相關性，且盡量保持原數據。
Matlab的一些知識：
函數句柄的好處就是把一個函數作為參數傳入到本函數中，在該函數內部可以利用該函數進行各種運算得出最后需要的結果，比如說函數中要用到各種求導求積分的方法，如果是傳入該函數經過各種運算后的值的話，那么在調用該函數前就需要不少代碼，這樣比較累贅，所以采用函數句柄后這些代碼直接放在了函數內部，每調用一次無需在函數外面實現那么多的東西。
Matlab中保存各種數據時可以采用save函數，並將其保持為.mat格式的，這樣在matlab的current folder中看到的是.mat格式的文件，但是直接在文件夾下看，它是不直接顯示后綴的，且顯示的是Microsoft Access Table Shortcut，也就是.mat的簡稱。
關於實驗的一些說明：
在Ng的教程和實驗中，它的輸入樣本矩陣是每一列代表一個樣本的，列數為樣本的總個數。
matlab中矩陣64*10w大小肯定是可以的。
在本次實驗中，ZCA Whitening是針對patches進行的，且patches的均值化是對每一維進行的（感覺這種均值化比較靠譜，前面有文章是進行對patch中一個樣本求均值，感覺那樣很不靠譜，不過那是在natural image中做的，因為natural image每一維的統計特性都一樣，所以可以那樣均值化，但還是感覺不太靠譜）。因為使用的是ZCA whitening，所以新的向量並沒有進行降維，只是去了相關性和讓每一維的方差都相等而已。另外，由此可見，在進行數據Whitening時並不需要對原始的大圖片進行whitening，而是你用什么數據輸入網絡去訓練就對什么數據進行whitening，而這里，是用的小patches來訓練的，所以應該對小patches進行whitening。
關於本次實驗的一些數據和變量分配如下：
總共需訓練的樣本矩陣大小為192*100000。因為輸入訓練的一個patch大小為8*8的，所以網絡的輸入層節點數為192（=8*8*3，因為是3通道的，每一列按照rgb的順序排列），另外本次試驗的隱含層個數為400，權值懲罰系數為0.003，稀疏性懲罰系數為5，稀疏性體現在3.5%的隱含層節點被激發。ZCA白化時分母加上0.1的值防止出現大的數值。
用的是Linear decoder，所以最后的輸出層的激發函數為1，即輸出和輸入相等。這樣在問題內部的計算量變小了點。
程序中最后需要把學習到的網絡權值給顯示出來，不過這個顯示的內容已經包括了whitening部分了，所以是whitening和sparse autoencoder的組合。程序中顯示用的是displayColorNetwork( (W*ZCAWhite)');
這里為什么要用(W*ZCAWhite)'呢？首先，使用W*ZCAWhite是因為每個樣本x輸入網絡，其輸出等價於W*ZCAWhite*x；另外，由於W*ZCAWhite的每一行才是一個隱含節點的變換值,而displayColorNetwork函數是把每一列顯示一個小圖像塊的，所以需要對其轉置。
實驗結果：
原始圖片截圖：
ZCA Whitening后截圖;
學習到的400個特征顯示如下：
實驗主要部分代碼：
%% CS294A/CS294W Linear Decoder Exercise % Instructions % ------------ % % This file contains code that helps you get started on the % linear decoder exericse. For this exercise, you will only need to modify % the code in sparseAutoencoderLinearCost.m. You will not need to modify % any code in this file. %%====================================================================== %% STEP 0: Initialization % Here we initialize some parameters used for the exercise. imageChannels = 3; % number of channels (rgb, so 3) patchDim = 8; % patch dimension numPatches = 100000; % number of patches visibleSize = patchDim * patchDim * imageChannels; % number of input units outputSize = visibleSize; % number of output units hiddenSize = 400; % number of hidden units %中間的隱含層還變多了 sparsityParam = 0.035; % desired average activation of the hidden units. lambda = 3e-3; % weight decay parameter beta = 5; % weight of sparsity penalty term epsilon = 0.1; % epsilon for ZCA whitening %%====================================================================== %% STEP 1: Create and modify sparseAutoencoderLinearCost.m to use a linear decoder, % and check gradients % You should copy sparseAutoencoderCost.m from your earlier exercise % and rename it to sparseAutoencoderLinearCost.m. % Then you need to rename the function from sparseAutoencoderCost to % sparseAutoencoderLinearCost, and modify it so that the sparse autoencoder % uses a linear decoder instead. Once that is done, you should check % your gradients to verify that they are correct. % NOTE: Modify sparseAutoencoderCost first! % To speed up gradient checking, we will use a reduced network and some % dummy patches debugHiddenSize = 5; debugvisibleSize = 8; patches = rand([8 10]);%隨機產生10個樣本，每個樣本為一個8維的列向量，元素值為0~1 theta = initializeParameters(debugHiddenSize, debugvisibleSize); [cost, grad] = sparseAutoencoderLinearCost(theta, debugvisibleSize, debugHiddenSize, ... lambda, sparsityParam, beta, ... patches); % Check gradients numGrad = computeNumericalGradient( @(x) sparseAutoencoderLinearCost(x, debugvisibleSize, debugHiddenSize, ... lambda, sparsityParam, beta, ... patches), theta); % Use this to visually compare the gradients side by side disp([numGrad cost]); diff = norm(numGrad-grad)/norm(numGrad+grad); % Should be small. In our implementation, these values are usually less than 1e-9. disp(diff); assert(diff < 1e-9, 'Difference too large. Check your gradient computation again'); % NOTE: Once your gradients check out, you should run step 0 again to % reinitialize the parameters %} %%====================================================================== %% STEP 2: Learn features on small patches % In this step, you will use your sparse autoencoder (which now uses a % linear decoder) to learn features on small patches sampled from related % images. %% STEP 2a: Load patches % In this step, we load 100k patches sampled from the STL10 dataset and % visualize them. Note that these patches have been scaled to [0,1] load stlSampledPatches.mat displayColorNetwork(patches(:, 1:100)); %% STEP 2b: Apply preprocessing % In this sub-step, we preprocess the sampled patches, in particular, % ZCA whitening them. % % In a later exercise on convolution and pooling, you will need to replicate % exactly the preprocessing steps you apply to these patches before % using the autoencoder to learn features on them. Hence, we will save the % ZCA whitening and mean image matrices together with the learned features % later on. % Subtract mean patch (hence zeroing the mean of the patches) meanPatch = mean(patches, 2); %注意這里減掉的是每一維屬性的均值，為什么會和其它的不同呢？ patches = bsxfun(@minus, patches, meanPatch);%每一維都均值化 % Apply ZCA whitening sigma = patches * patches' / numPatches; [u, s, v] = svd(sigma); ZCAWhite = u * diag(1 ./ sqrt(diag(s) + epsilon)) * u';%求出ZCAWhitening矩陣 patches = ZCAWhite * patches; figure displayColorNetwork(patches(:, 1:100)); %% STEP 2c: Learn features % You will now use your sparse autoencoder (with linear decoder) to learn % features on the preprocessed patches. This should take around 45 minutes. theta = initializeParameters(hiddenSize, visibleSize); % Use minFunc to minimize the function addpath minFunc/ options = struct; options.Method = 'lbfgs'; options.maxIter = 400; options.display = 'on'; [optTheta, cost] = minFunc( @(p) sparseAutoencoderLinearCost(p, ... visibleSize, hiddenSize, ... lambda, sparsityParam, ... beta, patches), ... theta, options);%注意它的參數 % Save the learned features and the preprocessing matrices for use in % the later exercise on convolution and pooling fprintf('Saving learned features and preprocessing matrices...\n'); save('STL10Features.mat', 'optTheta', 'ZCAWhite', 'meanPatch'); fprintf('Saved\n'); %% STEP 2d: Visualize learned features W = reshape(optTheta(1:visibleSize * hiddenSize), hiddenSize, visibleSize); b = optTheta(2*hiddenSize*visibleSize+1:2*hiddenSize*visibleSize+hiddenSize); figure; %這里為什么要用(W*ZCAWhite)'呢？首先，使用W*ZCAWhite是因為每個樣本x輸入網絡， %其輸出等價於W*ZCAWhite*x；另外，由於W*ZCAWhite的每一行才是一個隱含節點的變換值 %而displayColorNetwork函數是把每一列顯示一個小圖像塊的，所以需要對其轉置。 displayColorNetwork( (W*ZCAWhite)');
sparseAutoencoderLinearCost.m:
function [cost,grad] = sparseAutoencoderLinearCost(theta, visibleSize, hiddenSize, ... lambda, sparsityParam, beta, data) % -------------------- YOUR CODE HERE -------------------- % Instructions: % Copy sparseAutoencoderCost in sparseAutoencoderCost.m from your % earlier exercise onto this file, renaming the function to % sparseAutoencoderLinearCost, and changing the autoencoder to use a % linear decoder. % -------------------- YOUR CODE HERE -------------------- % The input theta is a vector because minFunc only deal with vectors. In % this step, we will convert theta to matrix format such that they follow % the notation in the lecture notes. 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); % Loss and gradient variables (your code needs to compute these values) m = size(data, 2);%樣本點的個數 %% ---------- YOUR CODE HERE -------------------------------------- % Instructions: Compute the loss for the Sparse Autoencoder and gradients % W1grad, W2grad, b1grad, b2grad % % Hint: 1) data(:,i) is the i-th example % 2) your computation of loss and gradients should match the size % above for loss, W1grad, W2grad, b1grad, b2grad % z2 = W1 * x + b1 % a2 = f(z2) % z3 = W2 * a2 + b2 % h_Wb = a3 = f(z3) z2 = W1 * data + repmat(b1, [1, m]); a2 = sigmoid(z2); z3 = W2 * a2 + repmat(b2, [1, m]); a3 = z3; rhohats = mean(a2,2); rho = sparsityParam; KLsum = sum(rho * log(rho ./ rhohats) + (1-rho) * log((1-rho) ./ (1-rhohats))); squares = (a3 - data).^2; squared_err_J = (1/2) * (1/m) * sum(squares(:)); weight_decay_J = (lambda/2) * (sum(W1(:).^2) + sum(W2(:).^2)); sparsity_J = beta * KLsum; cost = squared_err_J + weight_decay_J + sparsity_J;%損失函數值 % delta3 = -(data - a3) .* fprime(z3); % but fprime(z3) = a3 * (1-a3) delta3 = -(data - a3); beta_term = beta * (- rho ./ rhohats + (1-rho) ./ (1-rhohats)); delta2 = ((W2' * delta3) + repmat(beta_term, [1,m]) ) .* a2 .* (1-a2); W2grad = (1/m) * delta3 * a2' + lambda * W2; b2grad = (1/m) * sum(delta3, 2); W1grad = (1/m) * delta2 * data' + lambda * W1; b1grad = (1/m) * sum(delta2, 2); %------------------------------------------------------------------- % Convert weights and bias gradients to a compressed form % This step will concatenate and flatten all your gradients to a vector % which can be used in the optimization method. grad = [W1grad(:) ; W2grad(:) ; b1grad(:) ; b2grad(:)]; end %------------------------------------------------------------------- % We are giving you the sigmoid function, you may find this function % useful in your computation of the loss and the gradients. function sigm = sigmoid(x) sigm = 1 ./ (1 + exp(-x)); end
參考資料：
Deep learning：十七(Linear Decoders，Convolution和Pooling)
Exercise: Implement deep networks for digit classification