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 recently defined the Universal Method, which substantially generalizes all the known approaches including Strassen's Laser Method and Cohn and Umans' Group Theoretic Method. 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 $CW_q$ 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 $CW_q$ 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 achieved a bound of $\omega \leq 2.376$, by applying the Laser Method to $CW_q$. By our result, the fact that they did not achieve $\omega=2$ implies a lower bound on the Universal Method applied to $CW_q$. Indeed, if it were possible for the Universal Method applied to $CW_q$ to achieve $\omega=2$, then Coppersmith and Winograd's application of the Laser Method would have achieved $\omega=2$.