Limits on the Universal Method for Matrix Multiplication.

In this work, we prove limitations on the known methods for designing matrix multiplication algorithms. Alman and Vassilevska Williams [2] recently defined the Universal Method, which substantially generalizes all the known approaches including Strassen's Laser Method [20] and Cohn and Umans' Group Theoretic Method [9]. We prove concrete lower bounds on the algorithms one can design by applying the Universal Method to many different tensors. Our proofs use new tools for upper bounding the asymptotic slice rank of a wide range of tensors. Our main result is that the Universal method applied to any Coppersmith-Winograd tensor CWq cannot yield a bound on omega, the exponent of matrix multiplication, better than 2.16805. By comparison, it was previously only known that the weaker "Galactic Method" applied to CWq could not achieve an exponent of 2. We also study the Laser Method (which is, in principle, a highly special case of the Universal Method) and prove that it is "complete" for matrix multiplication algorithms: when it applies to a tensor T, it achieves omega = 2 if and only if it is possible for the Universal method applied to T to achieve omega = 2. Hence, the Laser Method, which was originally used as an algorithmic tool, can also be seen as a lower bounding tool. For example, in their landmark paper, Coppersmith and Winograd [12] achieved a bound of omega <= 2.376, by applying the Laser Method to CWq. By our result, the fact that they did not achieve omega = 2 implies a lower bound on the Universal Method applied to CWq. Indeed, if it were possible for the Universal Method applied to CWq to achieve omega = 2, then Coppersmith and Winograd's application of the Laser Method would have achieved omega = 2.