Matrix Multiplication with Less Arithmetic Complexity and IO Complexity

TL;DR

This paper introduces a new matrix multiplication algorithm that reduces both arithmetic and IO complexities, surpassing previous methods and breaking established IO complexity lower bounds for recursive Strassen-like algorithms.

Contribution

The paper presents a novel matrix multiplication algorithm that simultaneously reduces leading coefficients in arithmetic and IO complexities, improving upon prior Strassen-like algorithms.

Findings

Reduces leading coefficient in arithmetic complexity.

Reduces IO complexity below previous bounds.

Improves performance of Strassen-like algorithms.

Abstract

After Strassen presented the first sub-cubic matrix multiplication algorithm, many Strassen-like algorithms are presented. Most of them with low asymptotic cost have large hidden leading coefficient which are thus impractical. To reduce the leading coefficient, Cenk and Hasan give a general approach reducing the leading coefficient of $<2,2,2;7>$-algorithm to $5$ but increasing IO complexity. In 2017, Karstadt and Schwartz also reduce the leading coefficient of $<2,2,2;7>$-algorithm to $5$ by the Alternative Basis Matrix Multiplication method. Meanwhile, their method reduces the IO complexity and low-order monomials in arithmetic complexity. In 2019, Beniamini and Schwartz generalize Alternative Basis Matrix Multiplication method reducing leading coefficient in arithmetic complexity but increasing IO complexity. In this paper, we propose a new matrix multiplication algorithm which reduces leading coefficient both in arithmetic complexity and IO complexity. We apply our method to Strassen-like algorithms improving arithmetic complexity and IO complexity (the comparison with previous results are shown in Tables 1 and 2). Surprisingly, our IO complexity of $<3,3,3;23>$-algorithm is $14n^{\log_323}M^{-\frac{1}{2}} + o(n^{\log_323})$ which breaks Ballard's IO complexity low bound ($\Omega(n^{\log_323}M^{1-\frac{\log_323}{2}})$) for recursive Strassen-like algorithms.