Faster Rectangular Matrix Multiplication by Combination Loss Analysis

TL;DR

This paper extends a novel analysis technique for matrix multiplication exponents, originally applied to square matrices, to improve bounds on the exponent for rectangular matrix multiplication.

Contribution

It introduces a method to combine combination loss analysis with existing tensor power analysis for rectangular matrices, advancing the theoretical bounds.

Findings

Improved upper bounds on rectangular matrix multiplication exponent.

Unified analysis approach for square and rectangular matrix multiplication.

Enhanced understanding of tensor power techniques in matrix multiplication.

Abstract

Duan, Wu and Zhou (FOCS 2023) recently obtained the improved upper bound on the exponent of square matrix multiplication $\omega<2.3719$ by introducing a new approach to quantify and compensate the ``combination loss" in prior analyses of powers of the Coppersmith-Winograd tensor. In this paper we show how to use this new approach to improve the exponent of rectangular matrix multiplication as well. Our main technical contribution is showing how to combine this analysis of the combination loss and the analysis of the fourth power of the Coppersmith-Winograd tensor in the context of rectangular matrix multiplication developed by Le Gall and Urrutia (SODA 2018).