User:Fawly/Computational complexity of matrix multiplication: Difference between revisions
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The '''matrix multiplication exponent''', usually denoted {{math|ω}}, is the smallest real number for which any <math>n\times n</math> matrix over a field can be multiplied together using <math>n^{\omega + o(1)}</math> field operations. This notation is commonly used in [[algorithm]]s research, algorithms using matrix multiplication as a subroutine have meaningful bounds on running time regardless of the true value of {{math|ω}}.
Using a naive lower bound and schoolbook matrix multiplication for the upper bound, one can straightforwardly conclude that {{math|2 ≤ ω ≤ 3}}. Whether {{math|1=ω = 2}} is a major open question in [[theoretical computer science]], and there is a line of research developing matrix multiplication algorithms to get improved bounds on {{math|ω}}. The current best bound on {{math|ω}} is {{math|ω < 2.3728596}}, by Josh Alman and [[Virginia Vassilevska Williams]].<ref name="aw20"/> This algorithm, like all other recent algorithms in this line of research, uses the ''laser method'', a generalization of the Coppersmith–Winograd algorithm, which was given by [[Don Coppersmith]] and [[Shmuel Winograd]] in 1990, was the best matrix multiplication algorithm until 2010, and has an asymptotic complexity of {{math|''O''(''n''<sup>2.375477</sup>)}}.<ref name="coppersmith">{{Citation|doi=10.1016/S0747-7171(08)80013-2 |title=Matrix multiplication via arithmetic progressions |url=http://www.cs.umd.edu/~gasarch/TOPICS/ramsey/matrixmult.pdf |year=1990 |last1=Coppersmith |first1=Don |last2=Winograd |first2=Shmuel |journal=Journal of Symbolic Computation |volume=9|issue=3|pages=251}}</ref> The conceptual idea of these algorithms are similar to Strassen's algorithm: a way is devised for multiplying two {{math|''k'' × ''k''}}-matrices with fewer than {{math|''k''<sup>3</sup>}} multiplications, and this technique is applied recursively. The laser method has limitations to its power, and cannot be used to show that {{math|ω < 2.3725}}.<ref>{{Cite journal|last=Ambainis|first=Andris|last2=Filmus|first2=Yuval|last3=Le Gall|first3=François|date=2015-06-14|title=Fast Matrix Multiplication: Limitations of the Coppersmith-Winograd Method|url=https://doi.org/10.1145/2746539.2746554|journal=Proceedings of the forty-seventh annual ACM symposium on Theory of Computing|series=STOC '15|___location=Portland, Oregon, USA|publisher=Association for Computing Machinery|pages=585–593|doi=10.1145/2746539.2746554|isbn=978-1-4503-3536-2}}</ref>
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There is a trivial lower bound of {{tmath|\omega \ge 2}}. Since any algorithm for multiplying two {{math|''n'' × ''n''}}-matrices has to process all {{math|2''n''<sup>2</sup>}} entries, there is a trivial asymptotic lower bound of {{math|Ω(''n''<sup>2</sup>)}} operations for any matrix multiplication algorithm. Thus {{tmath|2\le \omega < 2.373}}. It is unknown whether {{tmath|\omega > 2}}. The best known lower bound for matrix-multiplication complexity is {{math|Ω(''n''<sup>2</sup> log(''n''))}}, for bounded coefficient [[Arithmetic circuit complexity|arithmetic circuits]] over the real or complex numbers, and is due to [[Ran Raz]].<ref>{{cite journal | last1 = Raz | first1 = Ran | author-link = Ran Raz | year = 2002| title = On the complexity of matrix product | journal = Proceedings of the Thirty-fourth Annual ACM Symposium on Theory of Computing | pages = 144 | doi = 10.1145/509907.509932 | isbn = 1581134959 | s2cid = 9582328 }}</ref>
==Related complexities==
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