On cap sets and the group-theoretic approach to matrix multiplication

Abstract

On cap sets and the group-theoretic approach to matrix multiplication, Discrete Analysis 2017:3, 27pp. A famous problem in computational complexity is to obtain a good estimate for the number of operations needed to compute the product of two $n\times n$ matrices. The obvious method uses $n^3$ operations, and it is initially tempting to think that one could not do substantially better. However, Strassen made a simple but very surprising observation that by cleverly grouping terms one can compute the product of two $2\times 2$ matrices using not eight but seven multiplications, and one can then iterate this idea to obtain an improved bound of $O(n^{\log 7/\log 2})$. This was the start of intensive research. The bound was improved to about 2.375 by Coppersmith and Winograd in 1990 and this was a natural barrier for various methods, but then it started to move again with improvements by Davie and Stothers and by Williams, with the current record of approximately 2.3728639 established by Le Gall in 2014 (to be compared with Williams's bound of 2.3728642). In the other direction, it is easy to show that at least $n^2$ operations are needed, since the product depends on all the matrix entries. The big problem is to determine whether the exponent 2 is the right one. Meanwhile, in 2003, Cohn and Umans had developed a new framework for thinking about the problem via group theory. A couple of years later, with Kleinberg and Szegedy, they used this approach to rederive the bound of Coppersmith and Winograd, and formulated some conjectures that would imply that the correct exponent was indeed 2 -- that is, that matrix multiplication can be performed using only $n^{2+o(1)}$ operations. Thanks to this work and later work by Umans with Alon and Shpilka, the matrix multiplication problem was found to be related to another famous open problem -- the Erdős-Szemerédi sunflower conjecture -- which in turn was related to yet another famous problem -- the cap set problem. As reported on the blog of this journal (and in many other places), the cap set problem was solved in a spectacular way by Ellenberg and Gijswijt, using a remarkable idea of Croot, Lev and Pach that had been used to prove a closely related result. This has had a knock-on effect on the other problems: it straight away proved a version of the sunflower conjecture, thereby ruling out a method proposed by Coppersmith and Winograd that would have shown that the exponent was 2. However, there were several other approaches arising out of the framework of Cohn and Umans that were not immediately ruled out. The main purpose of this paper is to show that a wide class of statements that would imply that the exponent is 2 are false. While this does not rule out proving that the exponent is 2 using the group-theoretic framework, it places significant restrictions on how such a proof could look, and is therefore an important advance in our understanding of this problem. One of the ingredients concerns the so-called _tri-coloured_ version of the cap-set problem. In the group $\mathbb F_3^n$, this is the following question. How many triples $(a_i,b_i,c_i)$ can there be in $\mathbb F_3^n$ if $a_i+b_i+c_i=0$ for every $i$, and moreover these are the only solutions to the equation $a_i+b_j+c_k=0$? Note that if $A$ is a subset of $\mathbb F_3^n$ that contains no non-trivial solutions to the equation $x+y+z=0$, then the triples $(x,x,x)$ with $x\in A$ satisfy the required property, so this is a generalization of the cap-set problem itself. A simple adaptation of the Ellenberg-Gijswijt proof yields bounds for this problem as well, with the same bounds as in the special case of cap-sets. One of the main results of this paper is to show that if some of the conjectures of Cohn, Kleinberg, Szegedy and Umans that would yield an exponent of 2 for matrix multiplication are true, then in the groups they concern one can find large lower bounds for the tricoloured version of the cap-set problem. It then goes on to show, using non-trivial generalizations of the techniques that work for $\mathbb F_3^n$, that for all Abelian groups of bounded exponent, such large lower bounds do not exist. It remains possible that an exponent of 2 for matrix multiplication can be established by following the group-theoretic framework, using Abelian groups of unbounded exponent or non-Abelian groups. This paper tells us that that is where we have to look.