Improved Rectangular Matrix Multiplication using Powers of the Coppersmith-Winograd Tensor

TL;DR

This paper extends the analysis of the Coppersmith-Winograd tensor to improve the asymptotic complexity bounds for rectangular matrix multiplication, leading to faster algorithms and better bounds on rectangular matrix product exponents.

Contribution

It generalizes tensor power analysis to asymmetric cases, improving bounds on rectangular matrix multiplication complexity and providing faster algorithms for various matrix shapes.

Findings

New lower bound for rectangular matrix multiplication exponent: α > 0.31389

Improved algorithms for multiplying n×n^k by n^k×n matrices for any k≠1

Enhanced understanding of tensor powers for asymmetric matrix multiplication

Abstract

In the past few years, successive improvements of the asymptotic complexity of square matrix multiplication have been obtained by developing novel methods to analyze the powers of the Coppersmith-Winograd tensor, a basic construction introduced thirty years ago. In this paper we show how to generalize this approach to make progress on the complexity of rectangular matrix multiplication as well, by developing a framework to analyze powers of tensors in an asymmetric way. By applying this methodology to the fourth power of the Coppersmith-Winograd tensor, we succeed in improving the complexity of rectangular matrix multiplication. Let $\alpha$ denote the maximum value such that the product of an $n\times n^\alpha$ matrix by an $n^\alpha\times n$ matrix can be computed with $O(n^{2+\epsilon})$ arithmetic operations for any $\epsilon>0$. By analyzing the fourth power of the Coppersmith-Winograd tensor using our methods, we obtain the new lower bound $\alpha>0.31389$, which improves the previous lower bound $\alpha>0.30298$ obtained five years ago by Le Gall (FOCS'12) from the analysis of the second power of the Coppersmith-Winograd tensor. More generally, we give faster algorithms computing the product of an $n\times n^k$ matrix by an $n^k\times n$ matrix for any value $k\neq 1$. (In the case $k=1$, we recover the bounds recently obtained for square matrix multiplication). These improvements immediately lead to improvements in the complexity of a multitude of fundamental problems for which the bottleneck is rectangular matrix multiplication, such as computing the all-pair shortest paths in directed graphs with bounded weights.