三层神经网络自编码算法推导和MATLAB实现 （转载）

%% CS294A/CS294W Programming Assignment Starter Code

%  Instructions
%  ------------
% 
%  This file contains code that helps you get started on the
%  programming assignment. You will need to complete the code in sampleIMAGES.m,
%  sparseAutoencoderCost.m and computeNumericalGradient.m. 
%  For the purpose of completing the assignment, you do not need to
%  change the code in this file. 
%
%%======================================================================
%% STEP 0: Here we provide the relevant parameters values that will
%  allow your sparse autoencoder to get good filters; you do not need to 
%  change the parameters below.

visibleSize = 8*8;   % number of input units 
hiddenSize = 25;     % number of hidden units 
sparsityParam = 0.01;   % desired average activation of the hidden units.
                     % (This was denoted by the Greek alphabet rho, which looks like a lower-case "p",
             %  in the lecture notes). 
lambda = 0.0001;     % weight decay parameter       
beta = 3;            % weight of sparsity penalty term       

%%======================================================================
%% STEP 1: Implement sampleIMAGES
%
%  After implementing sampleIMAGES, the display_network command should
%  display a random sample of 200 patches from the dataset

patches = sampleIMAGES;
display_network(patches(:,randi(size(patches,2),204,1)),8);%randi(size(patches,2),204,1)
                                                           %为产生一个204维的列向量，每一维的值为0~10000
                                                           %中的随机数，说明是随机取204个patch来显示


%  Obtain random parameters theta
theta = initializeParameters(hiddenSize, visibleSize);

%%======================================================================
%% STEP 2: Implement sparseAutoencoderCost
%
%  You can implement all of the components (squared error cost, weight decay term,
%  sparsity penalty) in the cost function at once, but it may be easier to do 
%  it step-by-step and run gradient checking (see STEP 3) after each step.  We 
%  suggest implementing the sparseAutoencoderCost function using the following steps:
%
%  (a) Implement forward propagation in your neural network, and implement the 
%      squared error term of the cost function.  Implement backpropagation to 
%      compute the derivatives.   Then (using lambda=beta=0), run Gradient Checking 
%      to verify that the calculations corresponding to the squared error cost 
%      term are correct.
%
%  (b) Add in the weight decay term (in both the cost function and the derivative
%      calculations), then re-run Gradient Checking to verify correctness. 
%
%  (c) Add in the sparsity penalty term, then re-run Gradient Checking to 
%      verify correctness.
%
%  Feel free to change the training settings when debugging your
%  code.  (For example, reducing the training set size or 
%  number of hidden units may make your code run faster; and setting beta 
%  and/or lambda to zero may be helpful for debugging.)  However, in your 
%  final submission of the visualized weights, please use parameters we 
%  gave in Step 0 above.

[cost, grad] = sparseAutoencoderCost(theta, visibleSize, hiddenSize, lambda, ...
                                     sparsityParam, beta, patches);

%%======================================================================
%% STEP 3: Gradient Checking
%
% Hint: If you are debugging your code, performing gradient checking on smaller models 
% and smaller training sets (e.g., using only 10 training examples and 1-2 hidden 
% units) may speed things up.

% First, lets make sure your numerical gradient computation is correct for a
% simple function.  After you have implemented computeNumericalGradient.m,
% run the following: 
checkNumericalGradient();

% Now we can use it to check your cost function and derivative calculations
% for the sparse autoencoder.  
numgrad = computeNumericalGradient( @(x) sparseAutoencoderCost(x, visibleSize, ...
                                                  hiddenSize, lambda, ...
                                                  sparsityParam, beta, ...
                                                  patches), theta);

% Use this to visually compare the gradients side by side
%disp([numgrad grad]); 

% Compare numerically computed gradients with the ones obtained from backpropagation
diff = norm(numgrad-grad)/norm(numgrad+grad);
disp(diff); % Should be small. In our implementation, these values are
            % usually less than 1e-9.

            % When you got this working, Congratulations!!! 

%%======================================================================
%% STEP 4: After verifying that your implementation of
%  sparseAutoencoderCost is correct, You can start training your sparse
%  autoencoder with minFunc (L-BFGS).

%  Randomly initialize the parameters
theta = initializeParameters(hiddenSize, visibleSize);

%  Use minFunc to minimize the function
addpath minFunc/
options.Method = ‘lbfgs‘; % Here, we use L-BFGS to optimize our cost
                          % function. Generally, for minFunc to work, you
                          % need a function pointer with two outputs: the
                          % function value and the gradient. In our problem,
                          % sparseAutoencoderCost.m satisfies this.
options.maxIter = 400;      % Maximum number of iterations of L-BFGS to run 
options.display = ‘on‘;


[opttheta, cost] = minFunc( @(p) sparseAutoencoderCost(p, ...
                                   visibleSize, hiddenSize, ...
                                   lambda, sparsityParam, ...
                                   beta, patches), ...
                              theta, options);

%%======================================================================
%% STEP 5: Visualization 

W1 = reshape(opttheta(1:hiddenSize*visibleSize), hiddenSize, visibleSize);
figure;
display_network(W1‘, 12); 

print -djpeg weights.jpg   % save the visualization to a file 

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 imageNum = 1:10%在每张图片中随机选取1000个patch，共10000个patch
    [rowNum colNum] = size(IMAGES(:,:,imageNum));
    for patchNum = 1:1000%实现每张图片选取1000个patch
        xPos = randi([1,rowNum-patchsize+1]);
        yPos = randi([1, colNum-patchsize+1]);
        patches(:,(imageNum-1)*1000+patchNum) = reshape(IMAGES(xPos:xPos+7,yPos:yPos+7,...
                                                        imageNum),64,1);
    end
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;%因为根据3sigma法则，95%以上的数据都在该区域内
                                                % 这里转换后将数据变到了-1到1之间

% Rescale from [-1,1] to [0.1,0.9]
patches = (patches + 1) * 0.4 + 0.1;

end

function theta = initializeParameters(hiddenSize, visibleSize)

%% Initialize parameters randomly based on layer sizes.
r  = sqrt(6) / sqrt(hiddenSize+visibleSize+1);   % we‘ll choose weights uniformly from the interval [-r, r]
W1 = rand(hiddenSize, visibleSize) * 2 * r - r;
W2 = rand(visibleSize, hiddenSize) * 2 * r - r;

b1 = zeros(hiddenSize, 1);
b2 = zeros(visibleSize, 1);

% Convert weights and bias gradients to the vector form.
% This step will "unroll" (flatten and concatenate together) all 
% your parameters into a vector, which can then be used with minFunc. 
theta = [W1(:) ; W2(:) ; b1(:) ; b2(:)];

end

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. 

%将长向量转换成每一层的权值矩阵和偏置向量值
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);

% 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. 
% 

Jcost = 0;%直接误差
Jweight = 0;%权值惩罚
Jsparse = 0;%稀疏性惩罚
[n m] = size(data);%m为样本的个数，n为样本的特征数

%前向算法计算各神经网络节点的线性组合值和active值
z2 = W1*data+repmat(b1,1,m);%注意这里一定要将b1向量复制扩展成m列的矩阵
a2 = sigmoid(z2);
z3 = W2*a2+repmat(b2,1,m);
a3 = sigmoid(z3);

% 计算预测产生的误差
Jcost = (0.5/m)*sum(sum((a3-data).^2));

%计算权值惩罚项
Jweight = (1/2)*(sum(sum(W1.^2))+sum(sum(W2.^2)));

%计算稀释性规则项
rho = (1/m).*sum(a2,2);%求出第一个隐含层的平均值向量
Jsparse = sum(sparsityParam.*log(sparsityParam./rho)+ ...
        (1-sparsityParam).*log((1-sparsityParam)./(1-rho)));

%损失函数的总表达式
cost = Jcost+lambda*Jweight+beta*Jsparse;

%反向算法求出每个节点的误差值
d3 = -(data-a3).*sigmoidInv(z3);
sterm = beta*(-sparsityParam./rho+(1-sparsityParam)./(1-rho));%因为加入了稀疏规则项，所以
                                                             %计算偏导时需要引入该项
d2 = (W2‘*d3+repmat(sterm,1,m)).*sigmoidInv(z2); 

%计算W1grad 
W1grad = W1grad+d2*data‘;
W1grad = (1/m)*W1grad+lambda*W1;

%计算W2grad  
W2grad = W2grad+d3*a2‘;
W2grad = (1/m).*W2grad+lambda*W2;

%计算b1grad 
b1grad = b1grad+sum(d2,2);
b1grad = (1/m)*b1grad;%注意b的偏导是一个向量，所以这里应该把每一行的值累加起来

%计算b2grad 
b2grad = b2grad+sum(d3,2);
b2grad = (1/m)*b2grad;



% %%方法二,每次处理1个样本，速度慢
% m=size(data,2);
% rho=zeros(size(b1));
% for i=1:m
%     %feedforward
%     a1=data(:,i);
%     z2=W1*a1+b1;
%     a2=sigmoid(z2);
%     z3=W2*a2+b2;
%     a3=sigmoid(z3);
%     %cost=cost+(a1-a3)‘*(a1-a3)*0.5;
%     rho=rho+a2;
% end
% rho=rho/m;
% sterm=beta*(-sparsityParam./rho+(1-sparsityParam)./(1-rho));
% %sterm=beta*2*rho;
% for i=1:m
%     %feedforward
%     a1=data(:,i);
%     z2=W1*a1+b1;
%     a2=sigmoid(z2);
%     z3=W2*a2+b2;
%     a3=sigmoid(z3);
%     cost=cost+(a1-a3)‘*(a1-a3)*0.5;
%     %backpropagation
%     delta3=(a3-a1).*a3.*(1-a3);
%     delta2=(W2‘*delta3+sterm).*a2.*(1-a2);
%     W2grad=W2grad+delta3*a2‘;
%     b2grad=b2grad+delta3;
%     W1grad=W1grad+delta2*a1‘;
%     b1grad=b1grad+delta2;
% end
% 
% kl=sparsityParam*log(sparsityParam./rho)+(1-sparsityParam)*log((1-sparsityParam)./(1-rho));
% %kl=rho.^2;
% cost=cost/m;
% cost=cost+sum(sum(W1.^2))*lambda/2.0+sum(sum(W2.^2))*lambda/2.0+beta*sum(kl);
% W2grad=W2grad./m+lambda*W2;
% b2grad=b2grad./m;
% W1grad=W1grad./m+lambda*W1;
% b1grad=b1grad./m;


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

%sigmoid函数的逆向求导函数
function sigmInv = sigmoidInv(x)

    sigmInv = sigmoid(x).*(1-sigmoid(x));
end

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 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. 

epsilon = 1e-4;
n = size(theta,1);
E = eye(n);
for i = 1:n
    delta = E(:,i)*epsilon;
    numgrad(i) = (J(theta+delta)-J(theta-delta))/(epsilon*2.0);
end

% n=size(theta,1);
% E=eye(n);
% epsilon=1e-4;
% for i=1:n
%     dtheta=E(:,i)*epsilon;
%     numgrad(i)=(J(theta+dtheta)-J(theta-dtheta))/epsilon/2.0;
% end

%% ---------------------------------------------------------------
end

function [] = checkNumericalGradient()
% This code can be used to check your numerical gradient implementation 
% in computeNumericalGradient.m
% It analytically evaluates the gradient of a very simple function called
% simpleQuadraticFunction (see below) and compares the result with your numerical
% solution. Your numerical gradient implementation is incorrect if
% your numerical solution deviates too much from the analytical solution.
  
% Evaluate the function and gradient at x = [4; 10]; (Here, x is a 2d vector.)
x = [4; 10];
[value, grad] = simpleQuadraticFunction(x);

% Use your code to numerically compute the gradient of simpleQuadraticFunction at x.
% (The notation "@simpleQuadraticFunction" denotes a pointer to a function.)
numgrad = computeNumericalGradient(@simpleQuadraticFunction, x);

% Visually examine the two gradient computations.  The two columns
% you get should be very similar. 
disp([numgrad grad]);
fprintf(‘The above two columns you get should be very similar.\n(Left-Your Numerical Gradient, Right-Analytical Gradient)\n\n‘);

% Evaluate the norm of the difference between two solutions.  
% If you have a correct implementation, and assuming you used EPSILON = 0.0001 
% in computeNumericalGradient.m, then diff below should be 2.1452e-12 
diff = norm(numgrad-grad)/norm(numgrad+grad);
disp(diff); 
fprintf(‘Norm of the difference between numerical and analytical gradient (should be < 1e-9)\n\n‘);
end


  
function [value,grad] = simpleQuadraticFunction(x)
% this function accepts a 2D vector as input. 
% Its outputs are:
%   value: h(x1, x2) = x1^2 + 3*x1*x2
%   grad: A 2x1 vector that gives the partial derivatives of h with respect to x1 and x2 
% Note that when we pass @simpleQuadraticFunction(x) to computeNumericalGradients, we‘re assuming
% that computeNumericalGradients will use only the first returned value of this function.

value = x(1)^2 + 3*x(1)*x(2);

grad = zeros(2, 1);
grad(1)  = 2*x(1) + 3*x(2);
grad(2)  = 3*x(1);

end

function [h, array] = display_network(A, opt_normalize, opt_graycolor, cols, opt_colmajor)
% This function visualizes filters in matrix A. Each column of A is a
% filter. We will reshape each column into a square image and visualizes
% on each cell of the visualization panel. 
% All other parameters are optional, usually you do not need to worry
% about it.
% opt_normalize: whether we need to normalize the filter so that all of
% them can have similar contrast. Default value is true.
% opt_graycolor: whether we use gray as the heat map. Default is true.
% cols: how many columns are there in the display. Default value is the
% squareroot of the number of columns in A.
% opt_colmajor: you can switch convention to row major for A. In that
% case, each row of A is a filter. Default value is false.
warning off all

%exist(A),测试A是否存在，‘var‘表示只检测变量
if ~exist(‘opt_normalize‘, ‘var‘) || isempty(opt_normalize)
    opt_normalize= true;
end

if ~exist(‘opt_graycolor‘, ‘var‘) || isempty(opt_graycolor)
    opt_graycolor= true;
end

if ~exist(‘opt_colmajor‘, ‘var‘) || isempty(opt_colmajor)
    opt_colmajor = false;
end

% rescale
A = A - mean(A(:));

%colormap(gray)表示用灰度场景
if opt_graycolor, colormap(gray); end

% compute rows, cols
[L M]=size(A);
sz=sqrt(L);
buf=1;
if ~exist(‘cols‘, ‘var‘)%没有给定列数的情况下
    if floor(sqrt(M))^2 ~= M %M不是平方数时
        n=ceil(sqrt(M));
        while mod(M, n)~=0 && n<1.2*sqrt(M), n=n+1; end
        m=ceil(M/n);%m是最终要的小patch图像的尺寸大小
    else
        n=sqrt(M);
        m=n;
    end
else
    n = cols;
    m = ceil(M/n);
end

array=-ones(buf+m*(sz+buf),buf+n*(sz+buf));

if ~opt_graycolor
    array = 0.1.* array;
end


if ~opt_colmajor
    k=1;
    for i=1:m
        for j=1:n
            if k>M, 
                continue; 
            end
            clim=max(abs(A(:,k)));
            if opt_normalize
                array(buf+(i-1)*(sz+buf)+(1:sz),buf+(j-1)*(sz+buf)+(1:sz))=reshape(A(:,k),sz,sz)/clim;
            else
                array(buf+(i-1)*(sz+buf)+(1:sz),buf+(j-1)*(sz+buf)+(1:sz))=reshape(A(:,k),sz,sz)/max(abs(A(:)));
            end
            k=k+1;
        end
    end
else
    k=1;
    for j=1:n
        for i=1:m
            if k>M, 
                continue; 
            end
            clim=max(abs(A(:,k)));
            if opt_normalize
                array(buf+(i-1)*(sz+buf)+(1:sz),buf+(j-1)*(sz+buf)+(1:sz))=reshape(A(:,k),sz,sz)/clim;
            else
                array(buf+(i-1)*(sz+buf)+(1:sz),buf+(j-1)*(sz+buf)+(1:sz))=reshape(A(:,k),sz,sz);
            end
            k=k+1;
        end
    end
end

if opt_graycolor
    h=imagesc(array,‘EraseMode‘,‘none‘,[-1 1]);%这里讲EraseMode设置为none,表示重绘时不擦除任何像素点
else
    h=imagesc(array,‘EraseMode‘,‘none‘,[-1 1]);
end
axis image off

drawnow;

warning on all

三层神经网络自编码算法推导和MATLAB实现 （转载）

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原文地址：http://www.cnblogs.com/jason-wyf/p/6849324.html