`svdsketch`

Low-rank SVD approximation chosen automatically to meet a tolerance.

Syntax

[U, S, V] = svdsketch(A)

Returns a truncated SVD whose rank is picked to reach the default tolerance sqrt(eps).

[U, S, V] = svdsketch(A, tol)

Uses the relative tolerance tol for norm(U*S*V' - A, 'fro')/norm(A, 'fro').

Description

You do not give the rank. The function grows the approximation until the relative error is below the tolerance, so a matrix with fast-decaying singular values yields a small k, and a loose tolerance yields an even smaller one.

It is intended for large matrices where an approximate answer is enough and a full svd is too costly.

Input arguments

A: Numeric matrix.
tol: Relative approximation tolerance, a scalar in (0, 1).

Output arguments

U, S, V: Factors of the approximation A ~ U*S*V', with the same number k of columns in U and V, and S a k-by-k diagonal matrix.

Example

Rank chosen by tolerance

A = magic(4);
[U, S, V] = svdsketch(A);
size(S)
norm(U*S*V' - A) < 1e-6

Output

ans = 1×2

     3     3
ans = logical
   1

svdsketch