Volume 17 (2021) Article 1 pp. 1-30
CCC 2019 Special Issue
Limits on the Universal Method for Matrix Multiplication
Revised: July 19, 2020
Published: September 3, 2021
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Keywords: matrix multiplication, algebraic complexity
ACM Classification: F.2.2
AMS Classification: 68Q17

Abstract: [Plain Text Version]

We prove limitations on the known methods for designing matrix multiplication algorithms. Alman and Vassilevska Williams (FOCS'18) recently defined the Universal Method, which generalizes all the known approaches, including Strassen's Laser Method (J. reine angew. Math., 1987) and Cohn and Umans's Group Theoretic Method (FOCS'03). 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 to give upper bounds on 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 $2.16805$. It was previously known (Alman and Vassilevska Williams, FOCS'18) that the weaker “Galactic Method” applied to $CW_q$ could not achieve an exponent of $2$.

We also study the Laser Method (which is a 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-bound tool for a large class of algorithms. For example, in their landmark paper, Coppersmith and Winograd (J. Symbolic Computation, 1990) 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 $CW_q$.

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A conference version of this paper appeared in the 34th Computational Complexity Conference, 2019.

Supported in part by NSF grants CCF-1651838 and CCF-1741615. The work reported here was done while the author was a Ph.D. student at MIT.