Convolutional neural networks(CNN) (五) PCA in 2D Exercise

{作为CNN学习入门的一部分，笔者在这里逐步给出UFLDL的各章节Exercise的个人代码实现，供大家参考指正}

理论部分可以在线参阅(页面最下方有中文选项)PCA到Implementing PCA/Whitening部分内容，

此次练习比较简单，只给出相应代码与结果：

pca_2d.m

close all

%%================================================================
%% Step 0: Load data
%  We have provided the code to load data from pcaData.txt into x.
%  x is a 2 * 45 matrix, where the kth column x(:,k) corresponds to
%  the kth data point.Here we provide the code to load natural image data into x.
%  You do not need to change the code below.

x = load('pcaData.txt','-ascii');
% figure(1);
% scatter(x(1, :), x(2, :),'r');
% title('Raw data');


%%================================================================
%% Step 1a: Implement PCA to obtain U 
%  Implement PCA to obtain the rotation matrix U, which is the eigenbasis
%  sigma. 

% -------------------- YOUR CODE HERE -------------------- 
%  u = zeros(size(x, 1)); % You need to compute this

%  You need to make sure that the data has been approximately zero-mean.
x = bsxfun(@minus, x, mean(x,2));
sigma = x * x' / size(x, 2);
[U,S,V] = svd(sigma);
u = U;

% -------------------------------------------------------- 
% hold on
% plot([0 u(1,1)], [0 u(2,1)]);
% plot([0 u(1,2)], [0 u(2,2)]);
% scatter(x(1, :), x(2, :), 'b', 'filled');
% hold off

%%================================================================
%% Step 1b: Compute xRot, the projection on to the eigenbasis
%  Now, compute xRot by projecting the data on to the basis defined
%  by U. Visualize the points by performing a scatter plot.

% -------------------- YOUR CODE HERE -------------------- 
%  xRot = zeros(size(x)); % You need to compute this
xRot = U' * x;          % rotated version of the data. 

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

% Visualise the covariance matrix. You should see a line across the
% diagonal against a blue background.
figure(2);
scatter(xRot(1, :), xRot(2, :));
title('xRot');

%%================================================================
%% Step 2: Reduce the number of dimensions from 2 to 1. 
%  Compute xRot again (this time projecting to 1 dimension).
%  Then, compute xHat by projecting the xRot back onto the original axes 
%  to see the effect of dimension reduction

% -------------------- YOUR CODE HERE -------------------- 
k = 1; % Use k = 1 and project the data onto the first eigenbasis
%  xHat = zeros(size(x)); % You need to compute this
xTilde = U(:,1:k)' * x; % reduced dimension representation of the data, 
                        % where k is the number of eigenvectors to keep
xHat = U(:,1:k) * xTilde; % projecting the xRot back onto the original axes

% -------------------------------------------------------- 
figure(3);
scatter(xHat(1, :), xHat(2, :));
title('xHat');


%%================================================================
%% Step 3: PCA Whitening
%  Complute xPCAWhite and plot the results.

epsilon = 1e-5;
% -------------------- YOUR CODE HERE -------------------- 
% xPCAWhite = zeros(size(x)); % You need to compute this

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

% -------------------------------------------------------- 
figure(4);
scatter(xPCAWhite(1, :), xPCAWhite(2, :));
title('xPCAWhite');

%%================================================================
%% Step 3: ZCA Whitening
%  Complute xZCAWhite and plot the results.

% -------------------- YOUR CODE HERE -------------------- 
%  xZCAWhite = zeros(size(x)); % You need to compute this
xZCAWhite = U * diag(1./sqrt(diag(S) + epsilon)) * U' * x;

% -------------------------------------------------------- 
figure(5);
scatter(xZCAWhite(1, :), xZCAWhite(2, :));
title('xZCAWhite');

%% Congratulations! When you have reached this point, you are done!
%  You can now move onto the next PCA exercise. :)
           

实验结果：

需要注意的是，在第一张图片中，实心圈点代表raw data而实心点代表zero-mean后的数据，之后的图也都是在zero-mean之后作出来的。