How to do canonical correlation analysis with regularization using matlab?

Kaho Chan

2016 12 24

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Hi, there,

I need to do CCA(canonical correlation analysis) with regularization between X (n*d1 matrix) and Y (n*d2 matrix). (X and Y is not full rank.)

The regularization is defined as following, with a relatively small lambda:

regularized S_xx = S_xx + lambda*eye(d1)
regularized S_yy = S_yy + lambda*eye(d2)

I was trying to use the built-in function canoncorr, but it seemed that this function does not support this feature. I read through the code but didn't find anywhere to do the edit as the built-in function used QR decomposition.

I need this as a golden version to compare with my own. Does anybody know how to solve this problem using Matlab? Thanks!

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Kaho Chan 2016-12-24

编辑：Kaho Chan 2016-12-24

在 MATLAB Online 中打开

I tried the following method, but it seemed that it didn't work well enough under 'Largest principal angle'.

 [Vx, Vy] = canoncorr(x, y);
 Vx = Vx(:, 1:k); % get the first k component
 Vy = Vy(:, 1:k);
 % tried to use svd
 sxx = x'*x/(n-1);
 syy = y'*y/(n-1);
 sxy = x'*y/(n-1);
 M = sxx^(-0.5) * sxy * syy^(-0.5); % d1*d2
 [Wx, ~, Wy] = svd(M);
 Wx = sxx^0.5*Wx;
 Wy = syy^0.5*Wy;
 Wx = Wx(:, 1:k); 
 Wy = Wy(:, 1:k);
 % =============== compare using 'Largest principal angle' =========
 cos_x = calc_cos(Wx, Vx, sxx);
 % it doesn't work well, only around 0.95 with n = 300, d1 =30, d2 = 30
 function cos_theta = calc_cos(W, V, B)
    W = GS(W,B); % Gram–Schmidt process under B-norm
    V = GS(V,B);
    cos_theta = min(svd(V'*B*W));
 end