New Bounds for Matrix Multiplication: from Alpha to Omega

TL;DR

This paper introduces an improved variant of the laser method, leading to tighter bounds on the matrix multiplication exponent and rectangular matrix multiplication, advancing theoretical limits in matrix algebra complexity.

Contribution

It develops a new laser method variant that improves bounds on the matrix multiplication exponent and rectangular matrix multiplication, building on recent techniques.

Findings

Improved bound on matrix multiplication exponent to ω ≤ 2.371552.

Enhanced bounds on rectangular matrix multiplication exponents.

Refined dual exponent α ≥ 0.321334.

Abstract

The main contribution of this paper is a new improved variant of the laser method for designing matrix multiplication algorithms. Building upon the recent techniques of [Duan, Wu, Zhou, FOCS 2023], the new method introduces several new ingredients that not only yield an improved bound on the matrix multiplication exponent $\omega$, but also improve the known bounds on rectangular matrix multiplication by [Le Gall and Urrutia, SODA 2018]. In particular, the new bound on $\omega$ is $\omega\le 2.371552$ (improved from $\omega\le 2.371866$). For the dual matrix multiplication exponent $\alpha$ defined as the largest $\alpha$ for which $\omega(1,\alpha,1)=2$, we obtain the improvement $\alpha \ge 0.321334$ (improved from $\alpha \ge 0.31389$). Similar improvements are obtained for various other exponents for multiplying rectangular matrices.