Faster Matrix Multiplication via Asymmetric Hashing

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 $\omega$. 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 it by using an asymmetric version of CW's hashing method. By analyzing the 8th power of the CW tensor, we give a new bound of $\omega<2.37188$, which improves the previous best bound of $\omega<2.37286$ [Alman & V.Williams 2020]. Our result breaks the lower bound of $2.3725$ in [Ambainis et al. 2014] because of the new method for analyzing component (constituent) tensors.