March 29, 2007
We describe certain special consequences of certain elementary methods from group theory for studying the algebraic complexity of matrix multiplication, as developed by H. Cohn, C. Umans et. al. in 2003 and 2005. The measure of complexity here is the exponent of matrix multiplication, a real parameter between 2 and 3, which has been conjectured to be 2. More specifically, a finite group may simultaneously "realize" several independent matrix multiplications via its regular algebra if it has a family of triples of "index" subsets which satisfy the so-called simultaneous triple product property (STPP), in which case the complexity of these several multiplications does not exceed the rank (complexity) of the algebra. This leads to bounds for the exponent in terms of the size of the group and the sizes of its STPP triples, as well as the dimensions of its distinct irreducible representations. Wreath products of Abelian with symmetric groups appear especially important, in this regard, and we give an example of such a group which shows that the exponent is less than 2.84, and could be possibly be as small as 2.02 depending on the number of simultaneous matrix multiplications it realizes.
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November 18, 2005
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 algebra...
September 8, 2007
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 mat...
January 28, 2011
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 $\omega$ 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.
July 24, 2003
We develop a new, group-theoretic approach to bounding the exponent of matrix multiplication. There are two components to this approach: (1) identifying groups G that admit a certain type of embedding of matrix multiplication into the group algebra C[G], and (2) controlling the dimensions of the irreducible representations of such groups. We present machinery and examples to support (1), including a proof that certain families of groups of order n^(2 + o(1)) support n-by-n ma...
April 8, 2022
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 $\omega = 2$, 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 $\omega=2$ within the group-...
August 25, 2009
We deduce some elementary pairwise disjointness and semi-disjointness conditions on triples of subsets in arbitrary groups satisfying the so-called triple product property (TPP) as originally defined by H. Cohn and C. Umans in 2003. This property TPP for a triple of group subsets, called a TPP triple, allows the group to "realize" matrix multiplication of dimensions the sizes of the subsets, with the subsets acting as indexing sets for input matrices which are embedded into t...
July 29, 2011
In the context of group-theoretic fast matrix multiplication the TPP capacity is used to bound the exponent $\omega$ of matrix multiplication. We prove a new and sharper upper bound for the TPP subgroup capacity of a finite group
May 1, 2013
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 mul...
April 27, 2011
In 2003 COHN and UMANS introduced a group-theoretic approach to fast matrix multiplication. This involves finding large subsets of a group $G$ satisfying the Triple Product Property (TPP) as a means to bound the exponent $\omega$ of matrix multiplication. We present two new characterizations of the TPP, which are useful for theoretical considerations and for TPP test algorithms. With this we describe all known TPP tests and implement them in GAP algorithms. We also compare th...
December 6, 2017
The Cohn-Umans group-theoretic approach to matrix multiplication suggests embedding matrix multiplication into group algebra multiplication, and bounding $\omega$ in terms of the representation theory of the host group. This framework is general enough to capture the best known upper bounds on $\omega$ and is conjectured to be powerful enough to prove $\omega = 2$, although finding a suitable group and constructing such an embedding has remained elusive. Recently it was shown...