A non-commutative algorithm for multiplying 4x4 matrices using 48 non-complex multiplications

2025-06-16

AI Summary

This paper addresses the long-standing open problem of determining the minimal number of scalar multiplications required for 4×4 matrix multiplication over the field of rational numbers. Prior constructions achieving 48 multiplications relied on extensions such as the complex field or fields of characteristic two. Method: We introduce a novel construction framework based on homogeneous group actions over ℚ, combined with basis transformations and linear program optimization. Contribution/Results: We present the first non-commutative bilinear algorithm for 4×4 matrix multiplication over ℚ requiring only 48 rational-coefficient scalar multiplications. The algorithm is valid over any ring containing 1/2 and establishes the current record for minimal multiplication count in the rational setting. Its asymptotic arithmetic complexity is (19/16)n^{2+log₂(3/2)} + o(n^{2+log₂(3/2)}), significantly broadening the algebraic scope of low-multiplication-complexity matrix multiplication algorithms.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Security and Privacy: Applications of cryptography

Abstract

The quest for non-commutative matrix multiplication algorithms in small dimensions has seen a lot of recent improvements recently. In particular, the number of scalar multiplications required to multiply two $4 imes4$ matrices was first reduced in cite{Fawzi:2022aa} from 49 (two recursion levels of Strassen's algorithm) to 47 but only in characteristic 2 or more recently to 48 incite{alphaevolve} but over complex numbers. We propose an algorithm in 48 multiplications with only rational coefficients, hence removing the complex number requirement. It was derived from the latter one, under the action of an isotropy which happen to project the algorithm on the field of rational numbers. We also produce a straight line program of this algorithm, reducing the leading constant in the complexity, as well as an alternative basis variant of it, leading to an algorithm running in ${frac{19}{16}n^{2+frac{log_2 3 }{2}} + oleft({n^{2+frac{log_2 3}{2}}} ight)}$ operations over any ring containing an inverse of 2.

Problem

• Reducing scalar multiplications for 4x4 matrix multiplication
• Eliminating complex number requirement in multiplication algorithm
• Optimizing algorithm complexity over any ring

Innovation

• 48 rational multiplications for 4x4 matrices
• Non-complex coefficients algorithm
• Improved complexity with straight line program