Towards Practical Fast Matrix Multiplication based on Trilinear Aggregation

Tor Hadas, Oded Schwartz

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

2 Scopus citations

Abstract

Pan's four decades old fast matrix multiplication algorithms have the lowest asymptotic complexity of all currently known algorithms applicable to matrices of feasible dimensions. However, the large coefficients in the arithmetic cost of these algorithms make them impractical. We reduce these coefficients by , in some cases to a value of 2, the same leading coefficient as the classical, cubic time algorithm. For this purpose, we utilize fast recursive transformations with sparsification of the linear operators of Pan's algorithms. Existing decomposition methods cannot be applied to Pan's algorithms due to their large base cases. We describe two new methods for finding such decompositions, by utilizing the underlying symmetries of the algorithms, and the linear transformations within the algorithms. With these tools, we obtain algorithms with the same asymptotic complexity as Pan's algorithms, but with small leading coefficients, often the same as that of the cubic time algorithm. In practice, these new algorithms have the potential to outperform the fastest currently known matrix multiplication algorithms on feasible sized inputs. Matched against known lower bounds, we show that our results are optimal or close to being optimal.

Original languageAmerican English
Title of host publicationISSAC 2023 - Proceedings of the 2023 International Symposium on Symbolic and Algebraic Computation
EditorsGabriela Jeronimo
PublisherAssociation for Computing Machinery
Pages289-297
Number of pages9
ISBN (Electronic)9798400700392
DOIs
StatePublished - 24 Jul 2023
Event48th International Symposium on Symbolic and Algebraic Computation, ISSAC 2023 - Tromso, Norway
Duration: 24 Jul 202327 Jul 2023

Publication series

NameACM International Conference Proceeding Series

Conference

Conference48th International Symposium on Symbolic and Algebraic Computation, ISSAC 2023
Country/TerritoryNorway
CityTromso
Period24/07/2327/07/23

Bibliographical note

Publisher Copyright:
© 2023 Owner/Author.

Keywords

  • Bilinear Algorithms
  • Fast Basis Transformation
  • Fast Recursive Transformation
  • Matrix Multiplication
  • Trilinear Aggregation

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