What's the truncation error in SVD?

Hi all,

I'm trying to prove one thing: perform SVD on matrix a, the truncation error of singular values equals to truncation error of products of singular vectors. Here is the code:

clear; clc;
% PART 1.
% define the matrix a. 
a = magic(8);
% define number of singular values left after truncation.
n = 2;
[u, s, v] = svd(a, 0);
% sum all singular values.
ds = diag(s);
dss = sum(ds);
% sum first n singular values
dssn = sum(ds(1:n));
% work out the error after truncation.
dsp = 1 - dssn / dss;

% PART 2
% multiply left singular vectors with singular values
u = u * s;
% select the first n singular vectors, i.e. to truncate
us = u(:, 1:n);
vs = v(:, 1:n);
% reconstruct the solution
ur = us * vs';
% work out the error
up = norm(ur - a, 'fro') / norm(a, 'fro');

I expect dsp = up, but dsp = 0.0431, up = 0.0613, why?

Answer 1

[Christine Tobler]

Use dsp = norm(ds(n+1:end)) / norm(ds) instead of the sum.

This is because in norm(U*S*V', 'fro'), you can move U and V outside of the norm since they are orthogonal, leaving you with norm(S, 'fro'), and since S is diagonal, this is equal to norm(diag(S)).