Fast Matrix Multiplication: Limitations of the Laser Method

Until a few years ago, the fastest known matrix multiplication algorithm, due to Coppersmith and Winograd (1990), ran in time O(n). 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(n), and identify a wide class of variants of this approach which cannot result in an algorithm with running time O(n); in particular, this approach cannot prove the conjecture that for every > 0, two n× n matrices can be multiplied in time O(n ). 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 is also the first to explain why taking tensor powers of the Coppersmith–Winograd identity results in faster algorithms. ∗Research done while visiting the Institute for Advanced Study.