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 <2. 371339, improved from the previous bound of <2. 371552. We also improve the bounds of the exponents for multiplying rectangular matrices of various shapes.