前言

實驗基礎說明：

1.為什麼要用線性解碼器，而不用前面用過的棧式自編碼器等？即：線性解碼器的作用？

這一點，Ng已經在講解中說明了，因為線性解碼器不用要求輸入資料範圍一定為(0,1），而前面用過的棧式自編碼器等要求輸入資料範圍必須為(0,1）。因為a3的輸出值是f函式的輸出，而在普通的sparse autoencoder中f函式一般為sigmoid函式，所以其輸出值的範圍為(0,1)，所以可以知道a3的輸出值範圍也在0到1之間。另外我們知道，在稀疏模型中的輸出層應該是儘量和輸入層特徵相同，也就是說a3=x1，這樣就可以推匯出x1也是在0和1之間，那就是要求我們對輸入到網路中的資料要先變換到0和1之間，這一條件雖然在有些領域滿足，比如前面實驗中的MINIST數字識別。但是有些領域，比如說使用了PCA Whitening後的資料，其範圍卻不一定在0和1之間。因此Linear Decoder方法就出現了。Linear Decoder是指在隱含層採用的激發函式是sigmoid函式，而在輸出層的激發函式採用的是線性函式，比如說最特別的線性函式——等值函式。

2.在實驗中，在ZCA whitening前進行資料預處理時，每列代表一個樣本，但為什麼是對patches的每行0均值化（即：每一維度0均值化，具體做法是：首先計算每一個維度上資料的均值（使用全體資料計算），之後在每一個維度上都減去該均值。），而以前的實驗都是對每列即每個樣本0均值化（即：逐樣本均值消減）？

①因為以前是灰度圖，現在是RGB彩色影象，如果現在對每列平均就是對三個通道求平均，這肯定不行。因為不同色彩通道中的畫素並不都存在平穩特性，而要進行逐樣本均值消減（即：單獨每個樣本0均值化）有一個必須滿足的前提：該資料是平穩的（見：資料預處理）。

②因為以前是自然影象，自然影象中畫素之間的統計特性都一樣，有一定的相關性，而現在是人工分割的影象塊，沒有這種特性。

3.在實驗中，把網路權值顯示出來為什麼是用displayColorNetwork( (W*ZCAWhite)')，而不像以前用的是display_Network( (W1)')?

 因為在本實驗中，資料patches在輸入網路前先經過了ZCA whitening的資料預處理，變成了ZCA白化後的資料ZCAWhite * patches，所以第一層隱含層輸出的實際上是W*ZCAWhite * patches，也就是說從原始資料patches到第一層隱含層輸出為W*ZCAWhite * patches的整個過程l轉換權值為W*ZCAWhite。

4.PCA Whitening和ZCA Whitening的區別

PCA Whitening：處理後的各資料方差都都相等，並都為1。主要用於降維和去相關性。

ZCA Whitening：處理後的各資料方差不一定為1，但一定相等。主要用於去相關性，且能儘量保持原始資料。

5.優秀的程式設計技巧：

要學會用函式控制代碼，比如patches = bsxfun(@minus, patches, meanPatch);

因為不使用函式控制代碼的情況下，對函式多次呼叫，每次都要為該函式進行全面的路徑搜尋，直接影響計算速度，藉助控制代碼可以完全避免這種時間損耗。也就是直接指定了函式的指標。函式控制代碼就像一個函式的名字，有點類似於C++程式中的引用。當然這一點已經在Deep Learning一之深度學習UFLDL教程：Sparse Autoencoder練習（斯坦福大學深度學習教程）中提到過，但我覺得有必須再強調一下。

實驗步驟

1.初始化引數，編寫計算線性解碼器代價函式及其梯度的函式sparseAutoencoderLinearCost.m，主要是在sparseAutoencoderCost.m的基礎上稍微修改，然後再檢查其梯度實現是否正確。

2.載入資料並原始資料進行ZCA Whitening的預處理。

3.學習特徵，即用LBFG演算法訓練整個線性解碼器網路，得到整個網路權值optTheta。

4.視覺化第一層學習到的特徵。

實驗結果

原始資料

ZCA Whitening後的資料

特徵視覺化結果，即：每一層學習到的特徵

 程式碼

linearDecoderExercise.m

%% 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 

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]);
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 grad]); 

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: 從pathes中學習特徵 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: 載入資料 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  %怎麼就就這個變數加到pathes上了呢？因為它裡面自己定義了變數patches的值！
figure;
displayColorNetwork(patches(:, 1:100)); 

%% STEP 2b: 預處理 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);  %為什麼是對每行求平均，以前是對每列即每個樣本求平均呀？因為以前是灰度圖，現在是彩色圖，如果現在對每列平均就是對三個通道求平均，這肯定不行
patches = bsxfun(@minus, patches, meanPatch);

% Apply ZCA whitening
sigma = patches * patches' / numPatches; %協方差矩陣
[u, s, v] = svd(sigma);
ZCAWhite = u * diag(1 ./ sqrt(diag(s) + epsilon)) * u';
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);

% 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;
displayColorNetwork( (W*ZCAWhite)');

sparseAutoencoderLinearCost.m

function [cost,grad,features] = sparseAutoencoderLinearCost(theta, visibleSize, hiddenSize, ...
                                                            lambda, sparsityParam, beta, data)
%計算線性解碼器代價函式及其梯度
% visibleSize:輸入層神經單元節點數   
% hiddenSize:隱藏層神經單元節點數  
% lambda: 權重衰減係數 
% sparsityParam: 稀疏性引數
% beta: 稀疏懲罰項的權重 
% data: 訓練集 
% theta：引數向量，包含W1、W2、b1、b2
% -------------------- 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); % 樣本數量

%% ---------- 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(:));              %均方差項
weight_decay_J = (lambda/2) * (sum(W1(:).^2) + sum(W2(:).^2));%權重衰減項
sparsity_J = beta * KLsum;                                    %懲罰項

cost = squared_err_J + weight_decay_J + sparsity_J;%損失函式值

% 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;   % W2梯度
b2grad = (1/m) * sum(delta3, 2);               % b2梯度
W1grad = (1/m) * delta2 * data' + lambda * W1; % W1梯度
b1grad = (1/m) * sum(delta2, 2);               % b1梯度

%-------------------------------------------------------------------
% 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

displayColorNetwork.m

function displayColorNetwork(A)

% display receptive field(s) or basis vector(s) for image patches
%
% A         the basis, with patches as column vectors

% In case the midpoint is not set at 0, we shift it dynamically
if min(A(:)) >= 0
    A = A - mean(A(:)); % 0均值化
end

cols = round(sqrt(size(A, 2)));% 每行大影象中小影象塊的個數

channel_size = size(A,1) / 3;
dim = sqrt(channel_size);   % 小影象塊內每行或列畫素點個數
dimp = dim+1;
rows = ceil(size(A,2)/cols);   % 每列大影象中小影象塊的個數
B = A(1:channel_size,:);                   % R通道畫素值
C = A(channel_size+1:channel_size*2,:);    % G通道畫素值
D = A(2*channel_size+1:channel_size*3,:);  % B通道畫素值
B=B./(ones(size(B,1),1)*max(abs(B)));% 歸一化
C=C./(ones(size(C,1),1)*max(abs(C)));
D=D./(ones(size(D,1),1)*max(abs(D)));
% Initialization of the image
I = ones(dim*rows+rows-1,dim*cols+cols-1,3);

%Transfer features to this image matrix
for i=0:rows-1
  for j=0:cols-1
      
    if i*cols+j+1 > size(B, 2)
        break
    end
    
    % This sets the patch
    I(i*dimp+1:i*dimp+dim,j*dimp+1:j*dimp+dim,1) = ...
         reshape(B(:,i*cols+j+1),[dim dim]);
    I(i*dimp+1:i*dimp+dim,j*dimp+1:j*dimp+dim,2) = ...
         reshape(C(:,i*cols+j+1),[dim dim]);
    I(i*dimp+1:i*dimp+dim,j*dimp+1:j*dimp+dim,3) = ...
         reshape(D(:,i*cols+j+1),[dim dim]);

  end
end

I = I + 1; % 使I的範圍從[-1,1]變為[0，2]
I = I / 2; % 使I的範圍從[0，2]變為[0, 1]
imagesc(I); 
axis equal  % 等比座標軸：設定螢幕高寬比，使得每個座標軸的具有均勻的刻度間隔
axis off    % 關閉所有的座標軸標籤、刻度、背景

end

參考資料

線性解碼器