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Group-Theoretic Lower Bounds for the Complexity of Matrix Multiplication

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Theory and Applications of Models of Computation (TAMC 2011)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6648))

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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 group-theoretic 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.

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Author information

Authors and Affiliations

  1. Computer Science Department, Saarland University, Campus E1.3, 66123, Saarbrücken, Germany

    Alexey Pospelov

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  1. Alexey Pospelov
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Editors and Affiliations

  1. Department of Computer Science, University of Miami, 1365 Memorial Drive, 33146, Coral Gables, FL, USA

    Mitsunori Ogihara

  2. Department of Information and Communication Engineering, University of Electro-Comm, Chofugaoga 1-5-1, Chofu, 182-8585, Tokyo, Japan

    Jun Tarui

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Pospelov, A. (2011). Group-Theoretic Lower Bounds for the Complexity of Matrix Multiplication. In: Ogihara, M., Tarui, J. (eds) Theory and Applications of Models of Computation. TAMC 2011. Lecture Notes in Computer Science, vol 6648. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-20877-5_2

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  • DOI: https://doi.org/10.1007/978-3-642-20877-5_2

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-20876-8

  • Online ISBN: 978-3-642-20877-5

  • eBook Packages: Computer ScienceComputer Science (R0)

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