9 research outputs found
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
Improved Bounds for Progression-Free Sets in C^n₈
Let G be a finite group, and let r₃(G) represent the size of the largest subset of G without non-trivial three-term progressions. In a recent breakthrough, Croot, Lev and Pach proved that r₃(C₄^n) ≤ (3.611)^n, where C_m denotes the cyclic group of order m. For finite abelian groups G≅∏^n_(i=1), where m₁,…,m_n denote positive integers such that m₁ |…|m_n, this also yields a bound of the form r₃(G)⩽(0.903)^(rk₄(G))|G|, with rk₄(G) representing the number of indices i ∈ {1,…, n} with 4 |m_i. In particular, r₃(Cn₈) ≤ (7.222)^n. In this paper, we provide an exponential improvement for this bound, namely r₃(Cn₈) ≤ (7.0899)^n
The asymptotic induced matching number of hypergraphs: balanced binary strings
We compute the asymptotic induced matching number of the -partite
-uniform hypergraphs whose edges are the -bit strings of Hamming weight
, for any large enough even number . Our lower bound relies on the
higher-order extension of the well-known Coppersmith-Winograd method from
algebraic complexity theory, which was proven by Christandl, Vrana and Zuiddam.
Our result is motivated by the study of the power of this method as well as of
the power of the Strassen support functionals (which provide upper bounds on
the asymptotic induced matching number), and the connections to questions in
tensor theory, quantum information theory and theoretical computer science.
Phrased in the language of tensors, as a direct consequence of our result, we
determine the asymptotic subrank of any tensor with support given by the
aforementioned hypergraphs. In the context of quantum information theory, our
result amounts to an asymptotically optimal -party stochastic local
operations and classical communication (slocc) protocol for the problem of
distilling GHZ-type entanglement from a subfamily of Dicke-type entanglement
Barriers for fast matrix multiplication from irreversibility
Determining the asymptotic algebraic complexity of matrix multiplication,
succinctly represented by the matrix multiplication exponent , is a
central problem in algebraic complexity theory. The best upper bounds on
, leading to the state-of-the-art , have been
obtained via the laser method of Strassen and its generalization by Coppersmith
and Winograd. Recent barrier results show limitations for these and related
approaches to improve the upper bound on .
We introduce a new and more general barrier, providing stronger limitations
than in previous work. Concretely, we introduce the notion of "irreversibility"
of a tensor and we prove (in some precise sense) that any approach that uses an
irreversible tensor in an intermediate step (e.g., as a starting tensor in the
laser method) cannot give . In quantitative terms, we prove that
the best upper bound achievable is lower bounded by two times the
irreversibility of the intermediate tensor. The quantum functionals and
Strassen support functionals give (so far, the best) lower bounds on
irreversibility. We provide lower bounds on the irreversibility of key
intermediate tensors, including the small and big Coppersmith--Winograd
tensors, that improve limitations shown in previous work. Finally, we discuss
barriers on the group-theoretic approach in terms of "monomial"
irreversibility
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 , 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 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' 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
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 in this potentially easier setting is a key challenge
that represents an intermediate goal short of actually proving . We
give two constructions in the continuous setting, each of which evades one of
our two barriers.Comment: 15 page
On Matrix Multiplication and Polynomial Identity Testing
We show that lower bounds on the border rank of matrix multiplication can be
used to non-trivially derandomize polynomial identity testing for small
algebraic circuits. Letting denote the border rank of matrix multiplication, we construct a hitting set generator
with seed length that hits
-variate circuits of multiplicative complexity . If the matrix
multiplication exponent is not 2, our generator has seed length
and hits circuits of size for
sufficiently small . Surprisingly, the fact that
already yields new, non-trivial hitting set
generators for circuits of sublinear multiplicative complexity