Computational complexity of matrix multiplication: Difference between revisions

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Similar techniques also apply to rectangular matrix multiplication. The central object of study is <math>\omega(k)</math>, which is the smallest <math>c</math> such that one can multiply a matrix of size <math>n\times \lceil n^k\rceil</math> with a matrix of size <math>\lceil n^k\rceil \times n</math> with <math>O(n^{c + o(1)})</math> arithmetic operations. A result in algebraic complexity states that multiplying matrices of size <math>n\times \lceil n^k\rceil</math> and <math>\lceil n^k\rceil \times n</math> requires the same number of arithmetic operations as multiplying matrices of size <math>n\times \lceil n^k\rceil</math> and <math>n \times n</math> and of size <math>n \times n</math> and <math>n\times \lceil n^k\rceil</math>, so this encompasses the complexity of rectangular matrix multiplication.<ref name="gall18">{{Citation|last1=Gall|first1=Francois Le|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|isbn=978-1-61197-503-1 |s2cid=33396059 }}</ref> This generalizes the square matrix multiplication exponent, since <math>\omega(1) = \omega</math>.

Since the output of the matrix multiplication problem is size <math>n^2</math>, we have <math>\omega(k) \geq 2</math> alwaysfor all values of <math>k</math>. If one can prove for some values of <math>k</math> between 0 and 1 that <math>\omega(k) \leq 2</math>, sothen thesesuch resultsa showresult shows that <math>\omega(k) = 2</math> exactlyfor those <math>k</math>. 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|last1=Cohen|first1=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|s2cid=231955576 }}</ref>

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|last1=Cohen|first1=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|s2cid=231955576 }}</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|last1=Le Gall|first1=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|isbn=978-1-61197-503-1 |s2cid=33396059 }}</ref> This paper also contains bounds on <math>\omega(k)</math>.