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ZCA白化变换推导——Learning Multiple Layers of Features from Tiny Images
参考文献:Learning Multiple Layers of Features from Tiny Images:附录
设数据集 X 的维数为 d×n ,且已经中心化
则协方差矩阵为
1/(n-1)*X*X‘
我们想让这n个d维向量中任意两维都不相关,则假定去相关矩阵为W
Y = W*X
为了使W达到去相关的目的,Y*Y‘必须是对角阵,可以进一步约束Y满足
Y * Y’ = (n - 1) I
再对W矩阵加限制(主要是方便下面的推导)
W = W‘
则
Y * Y’ = (n-1) I
W*X*X‘*W‘ = (n-1) I
W ‘ * W * X * X‘ * W = (n-1) * W‘ = (n-1) * W
所以W^2 * X * X‘ = (n-1) I
W = sqrt(n-1) * (X * X‘)^(-1/2)
而X * X‘ 是对称半正定,所以可以分解为 P*D*P’,其中D是对角阵,P是正交阵,(X * X‘)^(a) = P * D^(a) * P
所以W = P * D^(-1/2) * P‘;
Matlab代码
testZCA.m
clear;clc patches = []; tx = imread('test.jpg');%load('pcaData.txt','-ascii'); tx = double(tx); x = zeros(size(tx, 1) * size(tx, 2), size(tx, 3)); tx = tx(:); for i = 1 : size(x,2) x(:, i) = tx(1 + size(x, 1) * (i - 1) : size(x,1) * i); end patches = x'; %for i = 1 : size(x,2); % im = imread(strcat('train\', num2str(i), '.png')); % im = reshape(im, [1, 32*32*3]); % im = double(im); % % centralize % im(1:32*32) = im(1:32*32) - mean(im(1:32*32)); % im(32*32+1:2*32*32) = im(32*32+1:2*32*32) - mean(im(32*32+1:2*32*32)); % im(2*32*32+1:3*32*32) = im(2*32*32+1:3*32*32) - mean(im(2*32*32+1:3*32*32)); % % patches = [patches, im']; %end y = ZCA_whitening(patches); y = y * y'; y = ZCA_normalize(y); covx = 1/1000 * patches * patches'; covx = ZCA_normalize(covx); hold on subplot(1,2,1) imshow(uint8(covx)) subplot(1,2,2) imshow(uint8(y));
function y = ZCA_normalize(x) [row, col] = size(x); tx = []; for i = 1 : row tx = [tx, x(i,:)]; end tx = tx - min(tx); tx = tx / max(tx) * 255; y = zeros(row, col); for i = 1 : row y(i, :) = tx((i - 1) * col + 1 : i * col); end end
%% 假定每一个d维数据是0均值的,那么这n个d维数据的矩阵X(d*n)的协方差矩阵为 %% covX = 1/(n-1)*X*X' %% 为了消除维数之间的相关性,做变换W,得到Y,即Y = W*X %% 下面求‘去相关矩阵W'.由于W消除了相关性,所以Y*Y'是对角阵,故令W满足 %% Y*Y' = (n-1)*I %% 由于满足条件的W很多,那么不妨设W=W' %% 之后的推导比较自然,比较难想到的就是将X*X'做正交分解为P*D*P' function y = ZCAwhitening(x) [dim, n] = size(x); [P, D] = schur(x * x'); w = sqrt(n-1) * P * D^(-1/2) * P'; y = w * x; end效果图——数据集是取自Kaggle上的Tiny Image Classification的比赛
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