Group-theoretic Algorithms for Matrix Multiplication

@inproceedings{Cohn2005GrouptheoreticAF,
  title={Group-theoretic Algorithms for Matrix Multiplication},
  author={Henry Cohn and Robert D. Kleinberg and Bal{\'a}zs Szegedy and Christopher Umans},
  booktitle={FOCS},
  year={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 algebraic. Either one… 
Group-theoretic algorithms for matrix multiplication
  • C. Umans
  • Computer Science, Mathematics
    46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05)
  • 2005
TLDR
The group-theoretic approach to fast matrix multiplication introduced by Cohn and Umans is developed, and for the first time it is used to derive algorithms asymptotically faster than the standard algorithm.
Group-Theoretic Partial Matrix Multiplication
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
Group-Theoretic Lower Bounds for the Complexity of Matrix Multiplication
TLDR
Inspired by the recent group-theoretic approach by Cohn and Umans and the algorithms by Cohn et al. for matrix multiplication, conditional grouptheoretic lower bounds for the complexity of matrix multiplication are presented.
Analyzing group based matrix multiplication algorithms
TLDR
A variant of an algorithm based on the ideas exposed in [4] well-adapted for experimentation is introduced and it is shown how this approach can also be used for matrix multiplication over a field with characteristic different from 2.
The Simultaneous Triple Product Property and Group-theoretic Results for the Exponent of Matrix Multiplication
TLDR
Certain special consequences of certain elementary methods from group theory for studying the algebraic complexity of matrix multiplication are described, including wreath products of Abelian with symmetric groups, which could be possibly be as small as 2.02 depending on the number of simultaneous matrix multiplications it realizes.
Group-Theoretic Methods for bounding the exponent of matrix Multiplication
TLDR
It has been conjectured that w = 2, but the best known result is that w < 2.38, due to D. Coppersmith and S. Winograd.
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TLDR
This paper studies nonabelian groups as potential hosts for an embedding of matrix multiplication into group algebra multiplication and proves that symmetric groups cannot prove nontrivial bounds on $\omega$ when the embedding is via three Young subgroups.
Algorithmic problems in twisted groups of Lie type
TLDR
This thesis contains a collection of algorithms for working with the twisted groups of Lie type known as Suzuki groups, and small and large Ree groups, that have been developed with, and implemented in, the computer algebra system MAGMA.
Fast matrix multiplication using coherent configurations
TLDR
It is shown that bounds on ω can be established by embedding large matrix multiplication instances into small commutative coherent configurations, and a closure property involving symmetric powers of adjacency algebras is proved, which enables us to prove nontrivial bounds onπ� using commutATIVE coherent configurations and suggests that commutatives coherent configurations may be sufficient to prove ω = 2.
Fast matrix multiplication techniques based on the Adleman-Lipton model
TLDR
This discourse uses the tools of molecular biology to demonstrate the theoretical encoding of Strassen’s fast matrix multiplication algorithm with DNA based on an n-moduli set in the residue number system, thereby demonstrating the viability of computational mathematics with DNA.
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