More Asymmetry Yields Faster Matrix Multiplication

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. The method yields a new bound on the square matrix multiplication exponent ω<2.371339, improved from the previous bound of ω<2.371552. We also improve the bounds of the exponents for multiplying rectangular matrices of various shapes.