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Fast matrix multiplication using coherent configurations

Cohn, Henry and Umans, Christopher (2013) Fast matrix multiplication using coherent configurations. In: Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms. SIAM , Philadelphia, PA, pp. 1074-1087. ISBN 978-1-61197-251-1.

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We introduce a relaxation of the notion of tensor rank, called s-rank, and show that upper bounds on the s-rank of the matrix multiplication tensor imply upper bounds on the ordinary rank. In particular, if the “s-rank exponent of matrix multiplication” equals 2, then ω = 2. This connection between the s-rank exponent and the ordinary exponent enables us to significantly generalize the group-theoretic approach of Cohn and Umans, from group algebras to general algebras. Embedding matrix multiplication into general algebra multiplication yields bounds on s-rank (not ordinary rank) and, prior to this paper, that had been a barrier to working with general algebras. We identify adjacency algebras of coherent configurations as a promising family of algebras in the generalized framework. Coherent configurations are combinatorial objects that generalize groups and group actions; adjacency algebras are the analogue of group algebras and retain many of their important features. As with groups, coherent configurations support matrix multiplication when a natural combinatorial condition is satisfied, involving triangles of points in their underlying geometry. Finally, we prove a closure property involving symmetric powers of adjacency algebras, which enables us to prove nontrivial bounds on ω using commutative coherent configurations and suggests that commutative coherent configurations may be sufficient to prove ω = 2. Altogether, our results show that bounds on ω can be established by embedding large matrix multiplication instances into small commutative coherent configurations.

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Additional Information:© 2013 SIAM. [R]esearch supported by NSF grants CCF-0846991 and CCF-1116111 and BSF grant 2010120.
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Binational Science Foundation (USA-Israel)2010120
Record Number:CaltechAUTHORS:20160913-165513831
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ID Code:70326
Deposited On:29 Sep 2016 23:51
Last Modified:11 Nov 2021 04:27

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