TY - GEN
T1 - Fast matrix multiplication
T2 - 47th Annual ACM Symposium on Theory of Computing, STOC 2015
AU - Ambainis, Andris
AU - Filmus, Yuval
AU - Gall, François Le
PY - 2015/6/14
Y1 - 2015/6/14
N2 - Until a few years ago, the fastest known matrix multiplication algorithm, due to Coppersmith and Winograd (1990), ran in time O(n2.3755). Recently, a surge of activity by Stothers, Vassilevska-Williams, and Le Gall has led to an improved algorithm running in time O(n ' ). These algorithms are obtained by analyzing higher and higher tensor powers of a certain identity of Coppersmith and Winograd. We show that this exact approach cannot result in an algorithm with running time O(n2.3725), and identify a wide class of variants of this approach which cannot result in an algorithm with running time O(n2.3078); in particular, this approach cannot prove the conjecture that for every ∈ > 0, two n × n matrices can be multiplied in time O(n2+∈). We describe a new framework extending the original laser method, which is the method underlying the previously mentioned algorithms. Our framework accommodates the algorithms by Coppersmith and Winograd, Stothers, Vassilevska-Williams and Le Gall. We obtain our main result by analyzing this framework. The framework also explains why taking tensor powers of the Coppersmith-Winograd identity results in faster algorithms.
AB - Until a few years ago, the fastest known matrix multiplication algorithm, due to Coppersmith and Winograd (1990), ran in time O(n2.3755). Recently, a surge of activity by Stothers, Vassilevska-Williams, and Le Gall has led to an improved algorithm running in time O(n ' ). These algorithms are obtained by analyzing higher and higher tensor powers of a certain identity of Coppersmith and Winograd. We show that this exact approach cannot result in an algorithm with running time O(n2.3725), and identify a wide class of variants of this approach which cannot result in an algorithm with running time O(n2.3078); in particular, this approach cannot prove the conjecture that for every ∈ > 0, two n × n matrices can be multiplied in time O(n2+∈). We describe a new framework extending the original laser method, which is the method underlying the previously mentioned algorithms. Our framework accommodates the algorithms by Coppersmith and Winograd, Stothers, Vassilevska-Williams and Le Gall. We obtain our main result by analyzing this framework. The framework also explains why taking tensor powers of the Coppersmith-Winograd identity results in faster algorithms.
KW - Algebraic complexity theory
KW - Lower bounds
KW - Matrix multiplication
UR - http://www.scopus.com/inward/record.url?scp=84958750972&partnerID=8YFLogxK
U2 - 10.1145/2746539.2746554
DO - 10.1145/2746539.2746554
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AN - SCOPUS:84958750972
T3 - Proceedings of the Annual ACM Symposium on Theory of Computing
SP - 585
EP - 593
BT - STOC 2015 - Proceedings of the 2015 ACM Symposium on Theory of Computing
Y2 - 14 June 2015 through 17 June 2015
ER -