New Bounds for Matrix Multiplication: from Alpha to Omega

The main contribution of this paper is a new improved variant of the laser method for designing matrix multiplication algorithms. Building upon the recent techniques of [Duan, Wu, Zhou FOCS'2023], the new method introduces several new ingredients that not only yield an improved bound on the matrix multiplication exponent $\omega$, but also improves the known bounds on rectangular matrix multiplication by [Le Gall and Urrutia SODA'2018]. In particular, the new bound on $\omega$ is $\omega \le 2.371552$ (improved from $\omega \le 2.371866$). For the dual matrix multiplication exponent $\alpha$ defined as the largest $\alpha$ for which $\omega(1, \alpha, 1) = 2$, we obtain the improvement $\alpha \ge 0.321334$ (improved from $\alpha \ge 0.31389$). Similar improvements are obtained for various other exponents for multiplying rectangular matrices.