On the approximation of a matrix

TL;DR

This paper presents a method to improve matrix approximations by using a randomized algorithm to compute factors that outperform a given non-randomized approximation.

Contribution

It introduces a technique to enhance existing matrix approximations through randomized algorithms for factorization.

Findings

Randomized algorithms can produce better matrix approximations than non-randomized methods.

The method effectively computes factors H and T that improve the approximation of F.

The approach is applicable to various matrix approximation scenarios.

Abstract

Let $F^{*}$ be an approximation of a given $(a \times b)$ matrix $F$ derived by methods that are not randomized. We prove that for a given $F$ and $F^{*}$, $H$ and $T$ can be computed by randomized algorithm such that $(HT)$ is an approximation of $F$ better than $F^{*}$.