More Asymmetry Yields Faster Matrix Multiplication

TL;DR

This paper introduces a novel asymmetric approach to the laser method, leading to a tighter bound on the matrix multiplication exponent and improved bounds for rectangular matrix multiplication.

Contribution

It develops an asymmetric analysis technique that surpasses previous symmetric methods, achieving a new bound on the matrix multiplication exponent.

Findings

New bound on matrix multiplication exponent: ω<2.371339

Improved bounds for rectangular matrix multiplication

Advances in laser method with asymmetry

Abstract

We present a new improvement on the laser method for designing fast matrix multiplication algorithms. The new method further develops the recent advances by [Duan, Wu, Zhou FOCS 2023] and [Vassilevska Williams, Xu, Xu, Zhou SODA 2024]. Surprisingly the new improvement is achieved by incorporating more asymmetry in the analysis, circumventing a fundamental tool of prior work that requires two of the three dimensions to be treated identically. The method yields a new bound on the square matrix multiplication exponent $$\omega<2.371339,$$ improved from the previous bound of $\omega<2.371552$. We also improve the bounds of the exponents for multiplying rectangular matrices of various shapes.