How do I perform a truncated SVD on a matrix?

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David Haydock
David Haydock 2020-8-14
评论： David Haydock 2020-8-16
Title says it all. I have an input matrix that I need to generate a truncated SVD for, so that I can calculate the SVD's reconstruction error. I know that svd is a function, but I can't find any functions anywhere that work specifically for a truncated SVD.

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Bjorn Gustavsson
Bjorn Gustavsson 2020-8-14
Simply use the standard svd-function, then you can simply calculate the reconstruction-error at different truncation-levels by setting the eigenvalues outside of your trunkation to zero (that is what you do when trunkating, well close enough. If you want you can simply trunkate the U, S and V matrices too, but this way you cut out all thinking). Something like this:
Im = imread('cameraman.tif');
[U,S,V] = svd(double(Im));
subplot(1,3,1),imagesc(Im)
lambda = diag(S);
l50 = lambda;
l50(51:end) = 0;
subplot(1,3,2)
imagesc(U*diag(l50)*V')
subplot(1,3,3)
imagesc(double(Im)-U*diag(l50)*V')
HTH
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Bjorn Gustavsson
Bjorn Gustavsson 2020-8-16
For initial overview-reading I typically recommend wikipedia (SVD low-rank approximation) for getting the very first overview. Then you can branch out in your desired directions - I mainly use SVD for inverse-problems, your needs might take you in other directions.
Then, the truncated SVD is the "best lower-rank approximation" (minimum Frobenius-norm) of your original matrix. As for how that relates to conditional average is not clear to me. I've only ever encountered conditional averaging in the context of averaging time-serieses syncronized relative to some triggering event (that might occur at "random" instanses in time). To me that seems to be something completely different than a truncated SVD-approximation. I suggest going back to whomever told you this and ask for a more detailed explanation...

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