UFLDL教程练习（exercise）答案（2）

主成分分析与白化，这部分很简单，当然，其实是用Matlab比较简单，要是自己写SVD分解算法，足够研究好几个月的了。下面是我自己实现的练习答案，不保证完全正确，不过结果和网站上面给出的基本一致。

1.PCA in 2D

1.1 Step 1a: Implement PCA to obtain U

u = zeros(size(x, 1)); % You need to compute this
sigma = x * x' / size(x, 2);
[u,s,v]=svd(sigma);

1.2 Step 1b: Compute xRot, the projection on to the eigenbasis

xRot = zeros(size(x)); % You need to compute this
xRot=u'*x;

1.3 Step 2: Reduce the number of dimensions from 2 to 1. 

k = 1; % Use k = 1 and project the data onto the first eigenbasis
xHat = zeros(size(x)); % You need to compute this
x_ap=u(:,1:k)'*x;
xHat(1:k,:)=x_ap;
xHat=u*xHat;

1.4 Step 3: PCA Whitening

xPCAWhite = zeros(size(x)); % You need to compute this
xPCAWhite = diag(1./sqrt(diag(s) + epsilon)) * u' * x;

1.5 Step 3: ZCA Whitening

xZCAWhite = zeros(size(x)); % You need to compute this
xZCAWhite=u * diag(1./sqrt(diag(s) + epsilon)) * u' * x;

2.PCA and Whitening

2.1 Step 0b: Zero-mean the data (by row)

avg = mean(x, 1);
x = x - repmat(avg, size(x, 1), 1);

2.2 Step 1a: Implement PCA to obtain xRot

xRot = zeros(size(x)); % You need to compute this
sigma = x * x' / size(x, 2);
[U,S,V]=svd(sigma);
xRot=U'*x;

2.3 Step 1b: Check your implementation of PCA

covar = zeros(size(x, 1)); % You need to compute this
covar = xRot * xRot' / size(xRot, 2);

2.4 Step 2: Find k, the number of components to retain

k = 0; % Set k accordingly
sum_k=0;
sum=trace(S);
for k=1:size(S,1)
    sum_k=sum_k+S(k,k);
    if(sum_k/sum>=0.99) %0.9
        break;
    end
end

2.5 Step 3: Implement PCA with dimension reduction

xHat = zeros(size(x));  % You need to compute this
xTilde = U(:,1:k)' * x;
xHat(1:k,:)=xTilde;
xHat=U*xHat;

2.6 Step 4a: Implement PCA with whitening and regularisation

xPCAWhite = diag(1./sqrt(diag(S) + epsilon)) * U' * x;

2.7 Step 4b: Check your implementation of PCA whitening

covar = zeros(size(xPCAWhite, 1));
covar = xPCAWhite * xPCAWhite' / size(xPCAWhite, 2);

2.8 Step 5: Implement ZCA whitening

xZCAWhite=U * diag(1./sqrt(diag(S) + epsilon)) * U' * x;