Fast Approximate Matrix Multiplication by Solving Linear Systems

TL;DR

This paper introduces a deterministic algorithm for approximate matrix multiplication that efficiently computes an approximation with arbitrarily small Frobenius norm error in near-quadratic time by reducing the problem to solving linear systems.

Contribution

It presents the first deterministic quadratic-time algorithm for approximate matrix multiplication with guaranteed low absolute Frobenius norm error by reducing the problem to linear system solving.

Findings

Achieves approximate matrix multiplication in n^2 time.

Guarantees arbitrarily low Frobenius norm error.

First deterministic quadratic-time algorithm for this problem.

Abstract

In this paper, we present novel deterministic algorithms for multiplying two $n \times n$ matrices approximately. Given two matrices $A,B$ we return a matrix $C'$ which is an \emph{approximation} to $C = AB$. We consider the notion of approximate matrix multiplication in which the objective is to make the Frobenius norm of the error matrix $C-C'$ arbitrarily small. Our main contribution is to first reduce the matrix multiplication problem to solving a set of linear equations and then use standard techniques to find an approximate solution to that system in $\tilde{O}(n^2)$ time. To the best of our knowledge this the first examination into designing quadratic time deterministic algorithms for approximate matrix multiplication which guarantee arbitrarily low \emph{absolute error} w.r.t. Frobenius norm.