More Asymmetry Yields Faster Matrix Multiplication

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. Together with recent advances in numerical optimization [Dupont et al. arXiv:2608.16884], the method yields a new bound on the square matrix multiplication exponent $\omega <2.371177$, improved from the previous bound of $\omega <2.371552$. We also improve the bounds of the exponents for multiplying rectangular matrices of various shapes.

45 pages, in SODA 2025. arXiv admin note: text overlap with arXiv:2307.07970