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

@article{Blasiak2016OnCS, title={On cap sets and the group-theoretic approach to matrix multiplication}, author={Jonah Blasiak and Thomas Church and Henry Cohn and Joshua A. Grochow and Christopher Umans}, journal={ArXiv}, year={2016}, volume={abs/1605.06702} }

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…

## 94 Citations

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

- MathematicsArXiv
- 2017

This paper studies nonabelian groups as potential hosts for an embedding of matrix multiplication into group algebra multiplication and proves that symmetric groups cannot prove nontrivial bounds on $\omega$ when the embedding is via three Young subgroups.

Universal points in the asymptotic spectrum of tensors

- Mathematics, Computer ScienceSTOC
- 2018

It is proved that the quantum functionals are asymptotic upper bounds on slice-rank and multi-slice rank, extending a result of Tao and Sawin and extending the Coppersmith–Winograd method via combinatorial degeneration.

Slice Rank of Block Tensors and Irreversibility of Structure Tensors of Algebras

- Computer Science, MathematicsMFCS
- 2020

This work proves the following barrier over C: Starting with a tensor of minimal border rank satisfying a certain genericity condition, except for the diagonal tensor, it is impossible to prove ω = 2 using arbitrary restrictions.

Barriers for fast matrix multiplication from irreversibility

- Computer ScienceComputational Complexity Conference
- 2019

The notion of "irreversibility" of a tensor is introduced and it is proved that any approach that uses an irreversible tensor in an intermediate step cannot give ω = 2.37, proving that the best upper bound achievable is lower bounded by two times the irreversibility of the intermediate tensor.

Bounds for Matchings in Nonabelian Groups Will Sawin

- Mathematics
- 2018

We give upper bounds for triples of subsets of a finite group such that the triples of elements that multiply to 1 form a perfect matching. Our bounds are the first to give exponential savings in…

2 Barriers for Fast Matrix Multiplication from Irreversibility 1 Introduction 1 . 1 Matrix

- Computer Science
- 2019

The notion of “irreversibility” of a tensor is introduced and any approach that uses an irreversible tensor in an intermediate step cannot give ω = 2.37, proving that the best upper bound achievable is lower bounded by two times the irreversibility of the intermediate tensor.

Bounds for Matchings in Nonabelian Groups

- MathematicsElectron. J. Comb.
- 2018

Borders are the first to give exponential savings in powers of an arbitrary finite group such that the triples of elements that multiply to $1$ form a perfect matching.

Communication Complexity, Corner-Free Sets and the Symmetric Subrank of Tensors

- Mathematics, Computer ScienceArXiv
- 2021

It is proved that “Comon’s conjecture" about the equality of the rank and symmetric rank of symmetric tensors holds asymptotically for the tensor rank, the subrank as well as the restriction preorder.

Symmetric Subrank of Tensors and Applications

- Mathematics
- 2021

Strassen (Strassen, J. Reine Angew. Math., 375/376, 1987) introduced the subrank of a tensor as a natural extension of matrix rank to tensors. Subrank measures the largest diagonal tensor that can be…

## References

SHOWING 1-10 OF 47 REFERENCES

A group-theoretic approach to fast matrix multiplication

- Mathematics44th Annual IEEE Symposium on Foundations of Computer Science, 2003. Proceedings.
- 2003

A new, group-theoretic approach to bounding the exponent of matrix multiplication is developed, including a proof that certain families of groups of order n/sup 2+o(1)/ support n /spl times/ n matrix multiplication.

Group-theoretic algorithms for matrix multiplication

- Computer Science, Mathematics46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05)
- 2005

The group-theoretic approach to fast matrix multiplication introduced by Cohn and Umans is developed, and for the first time it is used to derive algorithms asymptotically faster than the standard algorithm.

Fast matrix multiplication using coherent configurations

- Mathematics, Computer ScienceSODA
- 2013

It is shown that bounds on ω can be established by embedding large matrix multiplication instances into small commutative coherent configurations, and a closure property involving symmetric powers of adjacency algebras is proved, which enables us to prove nontrivial bounds onπ� using commutATIVE coherent configurations and suggests that commutatives coherent configurations may be sufficient to prove ω = 2.

Geometric complexity theory and tensor rank

- MathematicsSTOC '11
- 2011

A very modest lower bound on the border rank of matrix multiplication tensors using G-representations is proved, which shows at least that the barrier for Gs-representation can be overcome.

Typical Tensorial Rank

- Mathematics
- 1997

The typical rank R(f) of a format f is the rank of Zariski almost all tensors of that format. Following Strassen [505] and Lickteig [331] we determine the asymptotic growth of the function R and…

Powers of tensors and fast matrix multiplication

- Computer ScienceISSAC
- 2014

This paper presents a method to analyze the powers of a given trilinear form and obtain upper bounds on the asymptotic complexity of matrix multiplication and obtains the upper bound ω < 2.3728639 on the exponent of square matrix multiplication, which slightly improves the best known upper bound.

Asymptotic upper bounds on progression-free sets in Zp

- Mathematics
- 2016

We show that any subset of Zp (p an odd prime) without 3-term arithmetic progression has size O(p), where c := 1 − 1 18 log p < 1. In particular, we find an upper bound of O(2.84) on the maximum size…

Partial and Total Matrix Multiplication

- Mathematics, Computer ScienceSIAM J. Comput.
- 1981

By combining Pan’s trilinear technique with a strong version of the compression theorem for the case of several disjoint matrix multiplications it is shown that multiplication of N \times N matrices (over arbitrary fields) is possible in time.

On the Asymptotic Complexity of Matrix Multiplication

- Computer Science, MathematicsSIAM J. Comput.
- 1982

A consequence of these results is that $\omega $, the exponent for matrix multiplication, is a limit point, that is, it cannot be realized by any single algorithm.

The growth rate of tri-colored sum-free sets

- MathematicsDiscrete Analysis
- 2018

The growth rate of tri-colored sum-free sets, Discrete Analysis 2018:12, 10 pp.
This paper contributes to the remarkable collection of results that followed in the wake of the 2016 breakthrough by…