Faster Matrix Multiplication via Asymmetric Hashing

作者：Ran Duan, Hongxun Wu, Renfei Zhou · 发表于：arXiv (Cornell University) · 年份：2022 · DOI：10.48550/arxiv.2210.10173 · 被引用次数：9 · 研究领域：Tensor decomposition and applications、Parallel Computing and Optimization Techniques

Fast matrix multiplication is one of the most fundamental problems in algorithm research. The exponent of the optimal time complexity of matrix multiplication is usually denoted by $ω$. This paper discusses new ideas for improving the laser method for fast matrix multiplication. We observe that the analysis of higher powers of the Coppersmith-Winograd tensor [Coppersmith & Winograd 1990] incurs a "combination loss", and we partially compensate for it using an asymmetric version of CW's hashing method. By analyzing the eighth power of the CW tensor, we give a new bound of $ω<2.371866$, which improves the previous best bound of $ω<2.372860$ [Alman & Vassilevska Williams 2020]. Our result breaks the lower bound of $2.3725$ in [Ambainis, Filmus & Le Gall 2015] because of the new method for analyzing component (constituent) tensors.