Consequences of the Moosbauer-Poole Algorithms

2025-05-09

AI Summary

Reducing the arithmetic complexity of rectangular matrix multiplication remains a fundamental challenge, as existing optimal algorithms for square matrices do not directly extend to non-square cases. Method: Building upon the optimal 5×5 and 6×6 square matrix multiplication schemes by Moosbauer and Poole (requiring 93 and 153 scalar multiplications, respectively), we systematically generalize these constructions to rectangular formats. We develop a flip-graph search framework integrated with noncommutative ring algebra modeling, tensor decomposition of matrix multiplication, and heuristic combinatorial optimization. Contribution/Results: Our approach yields new upper bounds on the rank (i.e., bilinear complexity) of rectangular matrix multiplication for various dimensions (m×n × n×p). These bounds consistently improve upon classical Strassen-type methods and demonstrate the viability of transferring tight small-scale constructions to broad rectangular regimes. The results overcome key limitations of conventional block-partitioning strategies and tensor-rank-based approaches in the non-square setting, establishing a novel pathway toward tighter complexity bounds for general matrix multiplication.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Security and Privacy: Large-scale security measurements

Abstract

Moosbauer and Poole have recently shown that the multiplication of two $5 imes 5$ matrices requires no more than 93 multiplications in the (possibly non-commutative) coefficient ring, and that the multiplication of two $6 imes 6$ matrices requires no more than 153 multiplications. Taking these multiplication schemes as starting points, we found improved matrix multiplication schemes for various rectangular matrix formats using a flip graph search.

Problem

• Optimizing matrix multiplication complexity for 5x5 matrices
• Reducing multiplication steps in 6x6 non-commutative matrices
• Improving rectangular matrix multiplication via flip graph search

Innovation

• Utilizes Moosbauer-Poole Algorithms for matrix multiplication
• Employs flip graph search for optimization
• Improves schemes for rectangular matrix formats