UFLDL Exercise:PCA and Whitening

這個練習還是圍繞着pca，pca白化，zca白化的，不過這裏用的圖像，而不再是簡單的二維數據，能夠讓我們直觀地看到這些預處理的作用

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

%% Step 0b: Zero-mean the data (by row)
%  You can make use of the mean and repmat/bsxfun functions.

% -------------------- YOUR CODE HERE -------------------- 
avg = mean(x,1);
x = x - repmat(avg,size(x,1),1);

Step 1a&1b: Implement PCA to obtain xRot & Check your implementation of PCA

%% Step 1a: Implement PCA to obtain xRot
%  Implement PCA to obtain xRot, the matrix in which the data is expressed
%  with respect to the eigenbasis of sigma, which is the matrix U.

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

%%================================================================
%% Step 1b: Check your implementation of PCA
%  The covariance matrix for the data expressed with respect to the basis U
%  should be a diagonal matrix with non-zero entries only along the main
%  diagonal. We will verify this here.
%  Write code to compute the covariance matrix, covar. 
%  When visualised as an image, you should see a straight line across the
%  diagonal (non-zero entries) against a blue background (zero entries).

% -------------------- YOUR CODE HERE -------------------- 
covar = zeros(size(x, 1)); % You need to compute this
covar = xRot * xRot' / size(xRot,2);
% Visualise the covariance matrix. You should see a line across the
% diagonal against a blue background.
figure('name','Visualisation of covariance matrix');
imagesc(covar);

結果如下

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

%% Step 2: Find k, the number of components to retain
%  Write code to determine k, the number of components to retain in order
%  to retain at least 99% of the variance.

% -------------------- YOUR CODE HERE -------------------- 
k = 0; % Set k accordingly
all = sum(diag(s));
for i=1:size(s,1)
    if sum(diag(s(1:i,1:i))) / all >= 0.99
        k = i;
        break;
    end
end

Step 3: Implement PCA with dimension reduction

%% Step 3: Implement PCA with dimension reduction
%  Now that you have found k, you can reduce the dimension of the data by
%  discarding the remaining dimensions. In this way, you can represent the
%  data in k dimensions instead of the original 144, which will save you
%  computational time when running learning algorithms on the reduced
%  representation.
% 
%  Following the dimension reduction, invert the PCA transformation to produce 
%  the matrix xHat, the dimension-reduced data with respect to the original basis.
%  Visualise the data and compare it to the raw data. You will observe that
%  there is little loss due to throwing away the principal components that
%  correspond to dimensions with low variation.

% -------------------- YOUR CODE HERE -------------------- 
xHat = zeros(size(x));  % You need to compute this
xTilde = u(:,1:k)' * x;
xHat = u(:,1:k) * xTilde;

% 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));

對比圖如下

原數據

保留90%的方差

保留99%的方差

Step 4a&4b: Implement PCA with whitening and regularisation & Check your implementation of PCA whitening

%% 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));
xPCAWhite = diag(sqrt(1./(diag(s) + epsilon))) * xRot;

% -------------------- YOUR CODE HERE -------------------- 

%%================================================================
%% Step 4b: Check your implementation of PCA whitening 
%  Check your implementation of PCA whitening with and without regularisation. 
%  PCA whitening without regularisation results a covariance matrix 
%  that is equal to the identity matrix. PCA whitening with regularisation
%  results in a covariance matrix with diagonal entries starting close to 
%  1 and gradually becoming smaller. We will verify these properties here.
%  Write code to compute the covariance matrix, covar. 
%
%  Without regularisation (set epsilon to 0 or close to 0), 
%  when visualised as an image, you should see a red line across the
%  diagonal (one entries) against a blue background (zero entries).
%  With regularisation, you should see a red line that slowly turns
%  blue across the diagonal, corresponding to the one entries slowly
%  becoming smaller.

% -------------------- YOUR CODE HERE -------------------- 

% Visualise the covariance matrix. You should see a red line across the
% diagonal against a blue background.
covar = xPCAWhite * xPCAWhite' / size(xPCAWhite,2)
figure('name','Visualisation of covariance matrix');
imagesc(covar);

epsilon = 0.1

epsilon = 0

Step 5: Implement ZCA whitening

%% Step 5: Implement ZCA whitening
%  Now implement ZCA whitening to produce the matrix xZCAWhite. 
%  Visualise the data and compare it to the raw data. You should observe
%  that whitening results in, among other things, enhanced edges.

xZCAWhite = zeros(size(x));

% -------------------- YOUR CODE HERE -------------------- 
xZCAWhite = u * xPCAWhite;
% Visualise the data, and compare it to the raw data.
% You should observe that the whitened images have enhanced edges.
figure('name','ZCA whitened images');
display_network(xZCAWhite(:,randsel));
figure('name','Raw images');
display_network(x(:,randsel));

結果如下，強化了邊緣