Related papers: Group-theoretic algorithms for matrix multiplicati…
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…
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 $\omega$…
We perform forward error analysis for a large class of recursive matrix multiplication algorithms in the spirit of [D. Bini and G. Lotti, Stability of fast algorithms for matrix multiplication, Numer. Math. 36 (1980), 63--72]. As a…
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 >…
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…
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…
The Cohn-Umans (FOCS '03) group-theoretic framework for matrix multiplication produces fast matrix multiplication algorithms from three subsets of a finite group $G$ satisfying a simple combinatorial condition (the Triple Product Property).…
A generalization of recent group-theoretic matrix multiplication algorithms to an analogue of the theory of partial matrix multiplication is presented. We demonstrate that the added flexibility of this approach can in some cases improve…
Cohn and Umans proposed a framework for developing fast matrix multiplication algorithms based on the embedding computation in certain groups algebras. In subsequent work with Kleinberg and Szegedy, they connected this to the search for…
In 2003, Cohn and Umans described a framework for proving upper bounds on the exponent $\omega$ of matrix multiplication by reducing matrix multiplication to group algebra multiplication, and in 2005 Cohn, Kleinberg, Szegedy, and Umans…
We consider the famous Strassen algorithm for fast multiplication of matrices. We show that this algorithm has a nontrivial finite group of automorphisms of order 36 (namely the direct product of two copies of the symmetric group on 3…
Group equivariant neural networks are growing in importance owing to their ability to generalise well in applications where the data has known underlying symmetries. Recent characterisations of a class of these networks that use high-order…
We show how to construct highly symmetric algorithms for matrix multiplication. In particular, we consider algorithms which decompose the matrix multiplication tensor into a sum of rank-1 tensors, where the decomposition itself consists of…
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…
We present a new improvement on the laser method for designing fast matrix multiplication algorithms. The new method further develops the recent advances by [Duan, Wu, Zhou FOCS 2023] and [Vassilevska Williams, Xu, Xu, Zhou SODA 2024].…
It is known that the multiplication of an $N \times M$ matrix with an $M \times P$ matrix can be performed using fewer multiplications than what the naive $NMP$ approach suggests. The most famous instance of this is Strassen's algorithm for…
We study the growth of product sets in some finite three-dimensional matrix groups. In particular, we prove two results about the group of $2\times 2$ upper triangular matrices over arbitrary finite fields: a product set estimate using…
We present a new algorithm for fast matrix multiplication using tensor decompositions which have special features. Thanks to these features we obtain exponents lower than what the rank of the tensor decomposition suggests. In particular for…
We study the algorithmic problem of multiplying large matrices that are rectangular. We prove that the method that has been used to construct the fastest algorithms for rectangular matrix multiplication cannot give algorithms with…
There have been several algorithms designed to optimise matrix multiplication. From schoolbook method with complexity $O(n^3)$ to advanced tensor-based tools with time complexity $O(n^{2.3728639})$ (lowest possible bound achieved), a lot of…