Close to optimal column approximations with a single SVD

TL;DR

This paper demonstrates that near-optimal column approximations in the Frobenius norm can be achieved with only a single SVD, significantly reducing computational costs compared to previous methods involving multiple SVDs or random sampling.

Contribution

It introduces a method to attain the same approximation bounds with just one SVD, simplifying and speeding up the process of finding high-quality column approximations and submatrices.

Findings

Achieves the Frobenius norm approximation bound with a single SVD.

Provides an efficient algorithm to find a nondegenerate submatrix in O(Nr^2) operations.

Reduces computational complexity compared to previous approaches.

Abstract

The best column approximation in the Frobenius norm with $r$ columns has an error at most $\sqrt{r+1}$ times larger than the truncated singular value decomposition. Reaching this bound in practice involves either expensive random volume sampling or at least $r$ executions of singular value decomposition. In this paper it will be shown that the same column approximation bound can be reached with only a single SVD (which can also be replaced with approximate SVD). As a corollary, it will be shown how to find a highly nondegenerate submatrix in $r$ rows of size $N$ in just $O(Nr^2)$ operations, which mostly has the same properties as the maximum volume submatrix.