A group-theoretic approach to fast matrix multiplication

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.

A Note on the Group-theoretic Approach to Fast Matrix Multiplication

In 2003 COHN and UMANS introduced a group-theoretic approach to fast matrix multiplication. This involves finding large subsets S, T and U of a group G satisfying the Triple Product Property (TPP) as a means to bound the exponent $ω$ of the matrix multiplication. We show that S, T and U may be be assumed to contain the identity and be otherwise disjoint. We also give a much shorter proof of the upper bound |S|+|T|+|U| <= |G|+2.

A Fast Search Algorithm for <m,m,m> Triple Product Property Triples and an Application for 5x5 Matrix Multiplication

We present a new fast search algorithm for <m,m,m> Triple Product Property (TPP) triples as defined by Cohn and Umans in 2003. The new algorithm achieves a speed-up factor of 40 up to 194 in comparison to the best known search algorithm. With a parallelized version of the new algorithm we are able to search for TPP triples in groups up to order 55. As an application we identify a list of groups that would realize 5x5 matrix multiplication with under 100 resp. 125 scalar multiplications (the best known upper bound by Makarov 1987 resp. the trivial upper bound) if they contain a <5,5,5> TPP trip…

Symmetries of matrix multiplication algorithms. I

In this work the algorithms of fast multiplication of matrices are considered. To any algorithm there associated a certain group of automorphisms. These automorphism groups are found for some well-known algorithms, including algorithms of Hopcroft, Laderman, and Pan. The automorphism group is isomorphic to $S_3\times Z_2$ and $S_4$ for Hopcroft anf Laderman algorithms, respectively. The studying of symmetry of algorithms may be a fruitful idea for finding fast algorithms, by an analogy with well-known optimization problems for codes, lattices, and graphs. {\em Keywords}: Strassen algorithm, sy…

快速矩阵乘法是稳定的

Fast matrix multiplication is stable

本文对一类递归矩阵乘法算法进行前向误差分析，证明矩阵乘法指数可通过数值稳定算法实现，并表明群论算法也属于稳定算法。

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 var…