A Refined Laser Method and Faster Matrix Multiplication

TL;DR

This paper introduces a refined laser method that improves the lower bounds on the matrix multiplication exponent , achieving the best known bound of <2.37286, and has potential for broader applications in arithmetic complexity.

Contribution

The paper presents a refined laser method that enhances the lower bounds on , leading to the best known upper bound for matrix multiplication complexity.

Findings

Achieved <2.37286, the best bound to date.

Refined the laser method to improve tensor value bounds.

Potential for further applications in arithmetic complexity.

Abstract

The complexity of matrix multiplication is measured in terms of $\omega$, the smallest real number such that two $n\times n$ matrices can be multiplied using $O(n^{\omega+\epsilon})$ field operations for all $\epsilon>0$; the best bound until now is $\omega<2.37287$ [Le Gall'14]. All bounds on $\omega$ since 1986 have been obtained using the so-called laser method, a way to lower-bound the `value' of a tensor in designing matrix multiplication algorithms. The main result of this paper is a refinement of the laser method that improves the resulting value bound for most sufficiently large tensors. Thus, even before computing any specific values, it is clear that we achieve an improved bound on $\omega$, and we indeed obtain the best bound on $\omega$ to date: $$\omega < 2.37286.$$ The improvement is of the same magnitude as the improvement that [Le Gall'14] obtained over the previous bound [Vassilevska W.'12]. Our improvement to the laser method is quite general, and we believe it will have further applications in arithmetic complexity.