Matrix multiplication algorithms from group orbits

Abstract

We show how to construct highly symmetric algorithms for matrix multiplication. In particular, we consider algorithms which decompose the matrix multiplication tensor into a sum of rank-1 tensors, where the decomposition itself consists of orbits under some finite group action. We show how to use the representation theory of the corresponding group to derive simple constraints on the decomposition, which we solve by hand for n = 2, 3, 4, 5, recovering Strassen's algorithm (in a particularly symmetric form) and new algorithms for larger n. While these new algorithms do not improve the known upper bounds on tensor rank or the matrix multiplication exponent, they are beautiful in their own right, and we point out modifications of this idea that could plausibly lead to further improvements. Our constructions also suggest further patterns that could be mined for new algorithms, including a tantalizing connection with lattices. In particular, using lattices we give the most transparent proof to date of Strassen's algorithm; the same proof works for all n, to yield a decomposition with n 3 − n + 1 terms.

Designing Strassen's algorithm

“…Although the method of construction suggested in [GM16], and independently in [CILO16], is more general than this, the constructions we ended up finding in [GM16] were in fact all instances of a single design-based construction yielding n 3 −n+1 multiplications for n×n matrix multiplication. The proof that this construction works is the simplest and most transparent proof of Strassen's algorithm that we are aware of.…”

“…This paper is a simplified and self-contained version of Section 5 of [GM16], in which we explored highly symmetric algorithms for multiplying matrices. Recently, there have been several papers analyzing the geometry and symmetries of algebraic algorithms for small matrices [Bur14, Bur15, LR16, LM16, CILO16].…”

“…Recently, there have been several papers analyzing the geometry and symmetries of algebraic algorithms for small matrices [Bur14, Bur15, LR16, LM16, CILO16]. In [GM16], we tried to take this line of research one step further by using symmetries to discover new algorithms for multiplying matrices of small size. While those algorithms did not improve the state-of-the-art bounds on the matrix multiplication exponent, they suggested that we can use group symmetries and group orbits to find new algorithms for matrix multiplication.…”

“…This leaves open the question of whether there are other families of highly symmetric algorithms. See [GM16,CILO16] for work in this direction, as well as [Bur14, Bur15, LR16, LM16, IL17].…”

Strassen's 2x2 matrix multiplication algorithm: A conceptual perspective

“…Clausen's construction is also describled in [7, Ch.1]. Grochow and Moore [16,17] generalize Clausen's construction to n × n matrices using other finite group orbits. Another symmetry is only apparent in the trilinear representation of the algorithm: the decompositions (⋆) are in one-to-one correspondence with decompositions of the trilinear form tr(XY Z) of the form…”