3,961 research outputs found
Powers of Tensors and Fast Matrix Multiplication
This paper presents a method to analyze the powers of a given trilinear form
(a special kind of algebraic constructions also called a tensor) and obtain
upper bounds on the asymptotic complexity of matrix multiplication. Compared
with existing approaches, this method is based on convex optimization, and thus
has polynomial-time complexity. As an application, we use this method to study
powers of the construction given by Coppersmith and Winograd [Journal of
Symbolic Computation, 1990] and obtain the upper bound on
the exponent of square matrix multiplication, which slightly improves the best
known upper bound.Comment: 28 page
Plethysm and fast matrix multiplication
Motivated by the symmetric version of matrix multiplication we study the
plethysm of the adjoint representation
of the Lie group . In particular, we describe the decomposition of this
representation into irreducible components for , and find highest weight
vectors for all irreducible components. Relations to fast matrix
multiplication, in particular the Coppersmith-Winograd tensor are presented.Comment: 5 page
Faster Algorithms for Rectangular Matrix Multiplication
Let {\alpha} be the maximal value such that the product of an n x n^{\alpha}
matrix by an n^{\alpha} x n matrix can be computed with n^{2+o(1)} arithmetic
operations. In this paper we show that \alpha>0.30298, which improves the
previous record \alpha>0.29462 by Coppersmith (Journal of Complexity, 1997).
More generally, we construct a new algorithm for multiplying an n x n^k matrix
by an n^k x n matrix, for any value k\neq 1. The complexity of this algorithm
is better than all known algorithms for rectangular matrix multiplication. In
the case of square matrix multiplication (i.e., for k=1), we recover exactly
the complexity of the algorithm by Coppersmith and Winograd (Journal of
Symbolic Computation, 1990).
These new upper bounds can be used to improve the time complexity of several
known algorithms that rely on rectangular matrix multiplication. For example,
we directly obtain a O(n^{2.5302})-time algorithm for the all-pairs shortest
paths problem over directed graphs with small integer weights, improving over
the O(n^{2.575})-time algorithm by Zwick (JACM 2002), and also improve the time
complexity of sparse square matrix multiplication.Comment: 37 pages; v2: some additions in the acknowledgment
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
Nondeterministic quantum communication complexity: the cyclic equality game and iterated matrix multiplication
We study nondeterministic multiparty quantum communication with a quantum
generalization of broadcasts. We show that, with number-in-hand classical
inputs, the communication complexity of a Boolean function in this
communication model equals the logarithm of the support rank of the
corresponding tensor, whereas the approximation complexity in this model equals
the logarithm of the border support rank. This characterisation allows us to
prove a log-rank conjecture posed by Villagra et al. for nondeterministic
multiparty quantum communication with message-passing.
The support rank characterization of the communication model connects quantum
communication complexity intimately to the theory of asymptotic entanglement
transformation and algebraic complexity theory. In this context, we introduce
the graphwise equality problem. For a cycle graph, the complexity of this
communication problem is closely related to the complexity of the computational
problem of multiplying matrices, or more precisely, it equals the logarithm of
the asymptotic support rank of the iterated matrix multiplication tensor. We
employ Strassen's laser method to show that asymptotically there exist
nontrivial protocols for every odd-player cyclic equality problem. We exhibit
an efficient protocol for the 5-player problem for small inputs, and we show
how Young flattenings yield nontrivial complexity lower bounds
Asymptotic tensor rank of graph tensors: beyond matrix multiplication
We present an upper bound on the exponent of the asymptotic behaviour of the
tensor rank of a family of tensors defined by the complete graph on
vertices. For , we show that the exponent per edge is at most 0.77,
outperforming the best known upper bound on the exponent per edge for matrix
multiplication (), which is approximately 0.79. We raise the question
whether for some the exponent per edge can be below , i.e. can
outperform matrix multiplication even if the matrix multiplication exponent
equals 2. In order to obtain our results, we generalise to higher order tensors
a result by Strassen on the asymptotic subrank of tight tensors and a result by
Coppersmith and Winograd on the asymptotic rank of matrix multiplication. Our
results have applications in entanglement theory and communication complexity
A note on the gap between rank and border rank
We study the tensor rank of the tensor corresponding to the algebra of
n-variate complex polynomials modulo the dth power of each variable. As a
result we find a sequence of tensors with a large gap between rank and border
rank, and thus a counterexample to a conjecture of Rhodes. At the same time we
obtain a new lower bound on the tensor rank of tensor powers of the generalised
W-state tensor. In addition, we exactly determine the tensor rank of the tensor
cube of the three-party W-state tensor, thus answering a question of Chen et
al.Comment: To appear in Linear Algebra and its Application
- …