42 research outputs found
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 Ο 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 Ο = 2. In this paper we rule out obtaining Ο = 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 matrices. The obvious method uses 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 matrices using not eight but seven multiplications, and one can then iterate this idea to obtain an improved bound of . 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 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 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 ErdΕs-SzemerΓ©di 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 , this is the following question. How many triples can there be in if for every , and moreover these are the only solutions to the equation ? Note that if is a subset of that contains no non-trivial solutions to the equation , then the triples with 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 , 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
Monochromatic Equilateral Triangles in the Unit Distance Graph
Let denote the minimum number of colors
needed to color so that there will not be a monochromatic
equilateral triangle with side length . Using the slice rank method, we
reprove a result of Frankl and Rodl, and show that
grows exponentially with . This
technique substantially improves upon the best known quantitative lower bounds
for , and we obtain Comment: 4 page
Upper bounds for sunflower-free sets
A collection of sets is said to form a -sunflower, or -system,
if the intersection of any two sets from the collection is the same, and we
call a family of sets sunflower-free if it contains no
sunflowers. Following the recent breakthrough of Ellenberg and Gijswijt and
Croot, Lev and Pach we apply the polynomial method directly to
Erd\H{o}s-Szemer\'{e}di sunflower problem and prove that any sunflower-free
family of subsets of has size at most We say that
a set for is
sunflower-free if every distinct triple there exists a coordinate
where exactly two of are equal. Using a version of the
polynomial method with characters
instead of polynomials, we
show that any sunflower-free set has size
where . This can be
seen as making further progress on a possible approach to proving the
Erd\H{o}s-Rado sunflower conjecture, which by the work of Alon, Sphilka and
Umans is equivalent to proving that for some constant
independent of .Comment: 5 page
The stability of finite sets in dyadic groups
We show that there is an absolute such that any subset of
of size is -stable in the sense of Terry
and Wolf. By contrast a size arithmetic progression in the integers is not
-stable.Comment: 9 pp; corrected some errors and expanded the introductio