Computational complexity of matrix multiplication: Difference between revisions

Revision as of 04:15, 28 May 2021

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Of interest is proving that, for values of ''k'' between 0 and 1, that <math>\omega(k) \leq 2</math>.

Since the output of the matrix multiplication problem is size <math>n^2</math>, <math>\omega(k) \geq 2</math> always, so these results show that <math>\omega(k) = 2</math> exactly. The largest ''k'' such that <math>\omega(k) = 2</math> is known as the ''dual matrix multiplication exponent'', usually denoted ''α''. ''α'' is referred to as the "[[Duality (optimization)|dual]]" because showing that <math>\alpha = 1</math> is equivalent to showing that <math>\omega = 2</math>. Like the matrix multiplication exponent, the dual matrix multiplication exponent sometimes appears in the complexity of algorithms in numerical linear algebra and optimization.<ref>{{Cite journal|last=Cohen|first=Michael B.|last2=Lee|first2=Yin Tat|last3=Song|first3=Zhao|date=2021-01-05|title=Solving Linear Programs in the Current Matrix Multiplication Time|url=https://doi.org/10.1145/3424305|journal=Journal of the ACM|volume=68|issue=1|pages=3:1–3:39|doi=10.1145/3424305|issn=0004-5411|arxiv=1810.07896}}</ref>

The first bound on ''α'' is by [[Don Coppersmith|Coppersmith]] in 1982, who showed that <math>\alpha > 0.17227</math>.<ref>{{Cite journal|last=Coppersmith|first=D.|date=1982-08-01|title=Rapid Multiplication of Rectangular Matrices|url=https://epubs.siam.org/doi/10.1137/0211037|journal=SIAM Journal on Computing|volume=11|issue=3|pages=467–471|doi=10.1137/0211037|issn=0097-5397}}</ref> The current best bound on ''α'' is <math>\alpha > 0.31389</math>, given by Le Gall and Urrutia.<ref>{{Citation|last=Le Gall|first=Francois|title=Improved Rectangular Matrix Multiplication using Powers of the Coppersmith-Winograd Tensor|date=2018-01-01|url=https://epubs.siam.org/doi/10.1137/1.9781611975031.67|work=Proceedings of the 2018 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)|pages=1029–1046|series=Proceedings|publisher=Society for Industrial and Applied Mathematics|doi=10.1137/1.9781611975031.67|access-date=2021-05-23|last2=Urrutia|first2=Florent|arxiv=1708.05622}}</ref> This paper also contains bounds on <math>\omega(k)</math>.