Skew-sparse matrix multiplication

TL;DR

This paper introduces a novel method for skew-sparse matrix multiplication over the rationals, leveraging algebraic structures to accelerate computations, with both deterministic and probabilistic algorithms tailored to the sparsity level.

Contribution

The paper develops a new algebraic approach for skew-sparse matrix multiplication over rationals, providing both deterministic and probabilistic algorithms with complexity improvements.

Findings

Deterministic algorithm with complexity $O(T^{ ext{ω}-2} p^2)$.

Probabilistic algorithm with complexity $O^ ilde{(t^{ ext{ω}-2} p^2)}$.

Method exploits isomorphism to quotient skew polynomial rings.

Abstract

Based on the observation that $\mathbb{Q}^{(p-1) \times (p-1)}$ is isomorphic to a quotient skew polynomial ring, we propose a new method for $(p-1)\times (p-1)$ matrix multiplication over $\mathbb{Q}$, where $p$ is a prime number. The main feature of our method is the acceleration for matrix multiplication if the product is skew-sparse. Based on the new method, we design a deterministic algorithm with complexity $O(T^{\omega-2} p^2)$, where $T\le p-1$ is a parameter determined by the skew-sparsity of input matrices and $\omega$ is the asymptotic exponent of matrix multiplication. Moreover, by introducing randomness, we also propose a probabilistic algorithm with complexity $O^\thicksim(t^{\omega-2}p^2+p^2\log\frac{1}{\nu})$, where $t\le p-1$ is the skew-sparsity of the product and $\nu$ is the probability parameter.