Group-theoretic lower bounds for the complexity of matrix multiplication

Authors:
Alexey Pospelov
Affiliations:
Saarland University, Computer Science Department, Saarbrücken, Germany
Venue:
TAMC'11 Proceedings of the 8th annual conference on Theory and applications of models of computation
Year:
2011

Citing 6
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Abstract

The complexity of multiplication in group algebras is closely related to the complexity of matrix multiplication. Inspired by the recent group-theoretic approach by Cohn and Umans [10] and the algorithms by Cohn et al. [9] for matrix multiplication, we present conditional grouptheoretic lower bounds for the complexity of matrix multiplication. These bounds depend on the complexity of multiplication in group algebras. Using Bläser's lower bounds for the rank of associative algebras we characterize all semisimple group algebras of minimal bilinear complexity and show improved lower bounds for other group algebras. We also improve the best previously known bound for the bilinear complexity of group algebras by Atkinson. Our bounds depend on the complexity of matrix multiplication. In the special if the exponent of the matrix multiplication equals two, we achieve almost linear bounds.