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

In 2003, Cohn and Umans described a framework for proving upper bounds on the exponent ω of matrix multiplication by reducing matrix multiplication to group algebra multiplication, and in 2005 Cohn, Kleinberg, Szegedy, and Umans proposed specific conjectures for how to obtain ω = 2. In this paper we rule out obtaining ω = 2 in this framework from abelian groups of bounded exponent. To do this we bound the size of tricolored sum-free sets in such groups, extending the breakthrough results of Croot, Lev, Pach, Ellenberg, and Gijswijt on cap sets. As a byproduct of our proof, we show that a variant of tensor rank due to Tao gives a quantitative understanding of the notion of unstable tensor from geometric invariant theory

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

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

Which groups are amenable to proving exponent two for matrix multiplication?

The Cohn-Umans group-theoretic approach to matrix multiplication suggests embedding matrix multiplication into group algebra multiplication, and bounding $\omega$ in terms of the representation theory of the host group. This framework is general enough to capture the best known upper bounds on $\omega$ and is conjectured to be powerful enough to prove $\omega = 2$, although finding a suitable group and constructing such an embedding has remained elusive. Recently it was shown, by a generalization of the proof of the Cap Set Conjecture, that abelian groups of bounded exponent cannot prove $\omega = 2$ in this framework, which ruled out a family of potential constructions in the literature. In this paper we study nonabelian groups as potential hosts for an embedding. We prove two main results: (1) We show that a large class of nonabelian groups---nilpotent groups of bounded exponent satisfying a mild additional condition---cannot prove $\omega = 2$ in this framework. We do this by showing that the shrinkage rate of powers of the augmentation ideal is similar to the shrinkage rate of the number of functions over $(\mathbb{Z}/p\mathbb{Z})^n$ that are degree $d$ polynomials; our proof technique can be seen as a generalization of the polynomial method used to resolve the Cap Set Conjecture. (2) We show that symmetric groups $S_n$ cannot prove nontrivial bounds on $\omega$ when the embedding is via three Young subgroups---subgroups of the form $S_{k_1} \times S_{k_2} \times \dotsb \times S_{k_\ell}$---which is a natural strategy that includes all known constructions in $S_n$. By developing techniques for negative results in this paper, we hope to catalyze a fruitful interplay between the search for constructions proving bounds on $\omega$ and methods for ruling them out.Comment: 23 pages, 1 figur

Group invariance principles for causal generative models

The postulate of independence of cause and mechanism (ICM) has recently led to several new causal discovery algorithms. The interpretation of independence and the way it is utilized, however, varies across these methods. Our aim in this paper is to propose a group theoretic framework for ICM to unify and generalize these approaches. In our setting, the cause-mechanism relationship is assessed by comparing it against a null hypothesis through the application of random generic group transformations. We show that the group theoretic view provides a very general tool to study the structure of data generating mechanisms with direct applications to machine learning.Comment: 16 pages, 6 figure

Uniquely Solvable Puzzles and Fast Matrix Multiplication

In 2003 Cohn and Umans introduced a new group-theoretic framework for doing fast matrix multiplications, with several conjectures that would imply the matrix multiplication exponent $\omega$ is 2. Their methods have been used to match one of the fastest known algorithms by Coppersmith and Winograd, which runs in $O(n^{2.376})$ time and implies that $\omega \leq 2.376$. This thesis discusses the framework that Cohn and Umans came up with and presents some new results in constructing combinatorial objects called uniquely solvable puzzles that were introduced in a 2005 follow-up paper, and which play a crucial role in one of the $\omega = 2$ conjectures

On generalized corners and matrix multiplication

Suppose that $S \subseteq [n]^2$ contains no three points of the form $(x,y), (x,y+\delta), (x+\delta,y')$, where $\delta \neq 0$. How big can $S$ be? Trivially, $n \le |S| \le n^2$. Slight improvements on these bounds are obtained from Shkredov's upper bound for the corners problem [Shk06], which shows that $|S| \le O(n^2/(\log \log n)^c)$ for some small $c > 0$, and a construction due to Petrov [Pet23], which shows that $|S| \ge \Omega(n \log n/\sqrt{\log \log n})$. Could it be that for all $\varepsilon > 0$, $|S| \le O(n^{1+\varepsilon})$? We show that if so, this would rule out obtaining $\omega = 2$ using a large family of abelian groups in the group-theoretic framework of Cohn, Kleinberg, Szegedy and Umans [CU03,CKSU05] (which is known to capture the best bounds on $\omega$ to date), for which no barriers are currently known. Furthermore, an upper bound of $O(n^{4/3 - \varepsilon})$ for any fixed $\varepsilon > 0$ would rule out a conjectured approach to obtain $\omega = 2$ of [CKSU05]. Along the way, we encounter several problems that have much stronger constraints and that would already have these implications.Comment: Feedback welcome