1,162 research outputs found
Group-theoretic algorithms for matrix multiplication
We further develop the group-theoretic approach to fast matrix multiplication
introduced by Cohn and Umans, and for the first time use it to derive
algorithms asymptotically faster than the standard algorithm. We describe
several families of wreath product groups that achieve matrix multiplication
exponent less than 3, the asymptotically fastest of which achieves exponent
2.41. We present two conjectures regarding specific improvements, one
combinatorial and the other algebraic. Either one would imply that the exponent
of matrix multiplication is 2.Comment: 10 page
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
in terms of the representation theory of the host group. This
framework is general enough to capture the best known upper bounds on
and is conjectured to be powerful enough to prove , 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
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 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 that are degree 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 cannot prove nontrivial bounds on
when the embedding is via three Young subgroups---subgroups of the
form ---which is a
natural strategy that includes all known constructions in .
By developing techniques for negative results in this paper, we hope to
catalyze a fruitful interplay between the search for constructions proving
bounds on and methods for ruling them out.Comment: 23 pages, 1 figur
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
Group-theoretic Methods for Bounding the Exponent of Matrix Multiplication
The (asymptotic) complexity of matrix multiplication (over the complex field)
is measured by a real parameter w > 0, called the exponent of matrix
multiplication (over the complex field), which is defined to be the smallest
real number w > 0 such that for an arbitrary degree of precision > 0, two n by
n complex matrices can be multiplied using an algorithm using O(n^(w+\epsilon))
number of non-division arithmetical operations. By the standard algorithm for
multiplying two matrices, the trivial lower and upper bounds for the exponent w
are 2 and 3 respectively.
W. Strassen in 1969 obtained the first important result that w < 2.81 using
his result that 2 by 2 matrix multiplication could be performed using 7
multiplications, not 8, as in the standard algorithm. In 1984, V. Pan improved
this to 2.67, using a variant of Strassen's approach. It has been conjectured
that w = 2, but the best known result is that w < 2.38, due to D. Coppersmith
and S. Winograd. In all these approaches, estimates for w depend on the number
of main running steps in their algorithms.
In a recent series of papers in 2003 and 2005, H. Cohn and C. Umans put
forward an entirely different approach using fairly elementary methods
involving finite groups, group algebras and their representations. The author
describes and proves their main results, and suggests possible ways of getting
improved estimates for the exponent using their methods.Comment: 96 page
Matrix Multiplication via Matrix Groups
In 2003, Cohn and Umans proposed a group-theoretic approach to bounding the exponent of matrix multiplication. Previous work within this approach ruled out certain families of groups as a route to obtaining ? = 2, while other families of groups remain potentially viable. In this paper we turn our attention to matrix groups, whose usefulness within this framework was relatively unexplored.
We first show that groups of Lie type cannot prove ? = 2 within the group-theoretic approach. This is based on a representation-theoretic argument that identifies the second-smallest dimension of an irreducible representation of a group as a key parameter that determines its viability in this framework. Our proof builds on Gowers\u27 result concerning product-free sets in quasirandom groups. We then give another barrier that rules out certain natural matrix group constructions that make use of subgroups that are far from being self-normalizing.
Our barrier results leave open several natural paths to obtain ? = 2 via matrix groups. To explore these routes we propose working in the continuous setting of Lie groups, in which we develop an analogous theory. Obtaining the analogue of ? = 2 in this potentially easier setting is a key challenge that represents an intermediate goal short of actually proving ? = 2. We give two constructions in the continuous setting, each of which evades one of our two barriers
On generalized corners and matrix multiplication
Suppose that contains no three points of the form , where . How big can be?
Trivially, . Slight improvements on these bounds are
obtained from Shkredov's upper bound for the corners problem [Shk06], which
shows that for some small , and a
construction due to Petrov [Pet23], which shows that .
Could it be that for all , ?
We show that if so, this would rule out obtaining 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
to date), for which no barriers are currently known. Furthermore, an
upper bound of for any fixed would
rule out a conjectured approach to obtain of [CKSU05]. Along the
way, we encounter several problems that have much stronger constraints and that
would already have these implications.Comment: Feedback welcome
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