A new quadratic-time number-theoretic algorithm to solve matrix multiplication problem

TL;DR

This paper introduces a quadratic-time number-theoretic algorithm for matrix multiplication that converts vectors into large entities to efficiently compute dot-products, offering a potentially faster approach than existing methods.

Contribution

The paper presents a novel quadratic-time algorithm for matrix multiplication based on number-theoretic techniques, differing from traditional tensor-based or recursive methods.

Findings

Quadratic time complexity with a significant constant factor.

Effective strategies for integers, floating point, and complex numbers.

Theoretical analysis of time and space complexity.

Abstract

There have been several algorithms designed to optimise matrix multiplication. From schoolbook method with complexity $O(n^3)$ to advanced tensor-based tools with time complexity $O(n^{2.3728639})$ (lowest possible bound achieved), a lot of work has been done to reduce the steps used in the recursive version. Some group-theoretic and computer algebraic estimations also conjecture the existence of an $O(n^2)$ algorithm. This article discusses a quadratic-time number-theoretic approach that converts large vectors in the operands to a single large entity and combines them to make the dot-product. For two $n \times n$ matrices, this dot-product is iteratively used for each such vector. Preprocessing and computation makes it a quadratic time algorithm with a considerable constant of proportionality. Special strategies for integers, floating point numbers and complex numbers are also discussed, with a theoretical estimation of time and space complexity.