I’m learning Prof. Andrew Ng’s Unsupervised Feature Learning and Deep Learning tutorial, This is part 2 of the 3rd exercise, which is use PCA algorithm in a natural image dataset. Just like my other UFLDL exercise posts, I’ll not go through the detail of the material. More details about this exercise can be found **HERE**.

### CODE

% pca_gen.m close all; clear all; %%================================================================ %% Step 0a: Load data % Here we provide the code to load natural image data into x. % x will be a 144 * 10000 matrix, where the kth column x(:, k) corresponds to % the raw image data from the kth 12x12 image patch sampled. % You do not need to change the code below. x = sampleIMAGESRAW(); figure('name','Raw images'); randsel = randi(size(x,2),200,1); % A random selection of samples for visualization display_network(x(:,randsel)); %%================================================================ %% Step 0b: Zero-mean the data (by row) % You can make use of the mean and repmat/bsxfun functions. % -------------------- YOUR CODE HERE -------------------- xmean = mean(x, 1); x = bsxfun(@minus, x, xmean); %%================================================================ %% 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 [nsamples, nfeatures] = size(x); sigma = x * x' ./ nfeatures; [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' ./ nfeatures; % Visualise the covariance matrix. You should see a line across the % diagonal against a blue background. figure('name','Visualisation of covariance matrix'); imagesc(covar); %%================================================================ %% 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 lambda = sum(S, 2); Sum = sum(lambda); temp = Sum; for i=size(lambda, 1):-1:1 lambda(i) temp = temp - lambda(i); if (temp / Sum < 0.99) k = i; break; end end %%================================================================ %% 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 xHat = U(:, 1:k) * U(:, 1:k)' * x; % 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)); % -------------------- YOUR CODE HERE -------------------- xPCAWhite = diag(1./sqrt(diag(S) + epsilon)) * U' * x; %%================================================================ %% 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 -------------------- covar = xPCAWhite * xPCAWhite' ./ nfeatures; % Visualise the covariance matrix. You should see a red line across the % diagonal against a blue background. figure('name','Visualisation of covariance matrix'); imagesc(covar); %%================================================================ %% 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)); |

### RESULTS

1. Raw image, which is randomly picked from the dataset.

2. Covariance matrix of xRot.

3.1. PCA result after determining k, which retains 99% of the variance. (116/144)

3.2. PCA result after determining k, which retains 70% of the variance. (13/144)

4. Covariance matrix after PCA Whiten.

5. ZCA whitened image.

🙂