文章标题：UFLDL：练习一

PCA and Whitening && Softmax Regression
(1)PCA

%%================================================================x = sampleIMAGESRAW();figure('name','Raw images');randsel = randi(size(x,2),200,1); % A random selection of samples for visualizationdisplay_network(x(:,randsel));%%================================================================avg = mean(x,2);x =  x - repmat(avg,1,size(x,2));%%================================================================xRot = zeros(size(x)); % You need to compute this[u,s,v]=svd(x);xRot = u'*x;%%================================================================covar = zeros(size(x, 1)); % You need to compute thiscovar = diag(diag(cov(x')));%%================================================================figure('name','Visualisation of covariance matrix');imagesc(covar);%%================================================================k = 0; % Set k accordinglyegis=eig(covar)egis=sort(egis,'descend')for i=1:size(covar,1)    if (sum(egis(1:i))/sum(egis)>0.99)         k=i        break;    endend%%================================================================%%================================================================xHat = zeros(size(x));  % You need to compute thisxHat = u*[xRot(1:k,:);zeros(size(xHat(k+1:end,:)))];% Visualise the data, and compare it to the raw data% You should observe that the raw and processed data are of comparable quality.% For comparison, you may wish to generate a PCA reduced image which% retains only 90% of the variance.figure('name',['PCA processed images ',sprintf('(%d / %d dimensions)', k, size(x, 1)),'']);display_network(xHat(:,randsel));figure('name','Raw images');display_network(x(:,randsel));%%================================================================%% Step 4a: Implement PCA with whitening and regularisation%  Implement PCA with whitening and regularisation to produce the matrix%  xPCAWhite. epsilon = 0.1;xPCAWhite = zeros(size(x));avg = mean(x, 1);     % Compute the mean pixel intensity value separately for each patch. x = x - repmat(avg, size(x, 1), 1);sigma = x * x' / size(x, 2);[U,S,V] = svd(sigma);xRot = U' * x;          % rotated version of the data. xTilde = U(:,1:k)' * x; % reduced dimension representation of the data,                         % where k is the number of eigenvectors to keepxPCAWhite = diag(1./sqrt(diag(S) + epsilon)) * U' * x;%%================================================================% Visualise the covariance matrix. You should see a red line across the% diagonal against a blue background.covar = diag(diag(cov(xPCAWhite')));figure('name','Visualisation of covariance matrix');imagesc(covar);%%================================================================xZCAWhite = zeros(size(x));xZCAWhite = U * diag(1./sqrt(diag(S) + epsilon)) * U' * x;%%================================================================figure('name','ZCA whitened images');display_network(xZCAWhite(:,randsel));figure('name','Raw images');display_network(x(:,randsel));

(2) Softmax Regression

function [cost, grad] = softmaxCost(theta, numClasses, inputSize, lambda, data, labels)% numClasses - the number of classes % inputSize - the size N of the input vector% lambda - weight decay parameter% data - the N x M input matrix, where each column data(:, i) corresponds to%        a single test set% labels - an M x 1 matrix containing the labels corresponding for the input data%% Unroll the parameters from thetatheta = reshape(theta, numClasses, inputSize);numCases = size(data, 2);groundTruth = full(sparse(labels, 1:numCases, 1));  %numClasses*Mcost = 0;thetagrad = zeros(numClasses, inputSize);M = theta*data;     % (numClasses,N)*(N,M)M = bsxfun(@minus, M, max(M, [], 1));h = exp(M);h =  bsxfun(@rdivide, h, sum(h));cost = -1/numCases*sum(sum(groundTruth.*log(h)))+lambda/2*sum(sum(theta.^2));thetagrad = -1/numCases*((groundTruth-h)*data')+lambda*theta;%log(h)下面一段是关键部分没有Vectorization版本的代码%for i=1:numCases%       s=groundTruth(:,i).*log(h(:,i));%      cost=cost+sum(s);%end%cost=cost*(-1)/numCases+lambda/2*sum(sum(theta.^2));%for i=1:numClasses%    for j=1:numCases%        %groundTruth(:,j)%        %h(:,j)%        k=((groundTruth(:,j)-h(:,j))*data(:,j)');%        %        thetagrad(i,:)=thetagrad(i,:)+k(i,:);%    end%     thetagrad(i,:)=-thetagrad(i,:)/numCases+lambda*theta(i,:);%endgrad = [thetagrad(:)];end