Exploring the Meta Flip Graph for Matrix Multiplication

2025-10-22

AI Summary

This work addresses the problem of optimizing upper bounds on the tensor rank of matrix multiplication. We propose a novel structural analysis framework based on the *meta-flip graph*, integrating flip graph theory, combinatorial optimization, and algebraic complexity analysis to enable fine-grained structural decomposition and rank estimation of tensors. Systematically applied to approximately thirty classical matrix multiplication formats, our framework improves the best-known tensor rank upper bounds for the vast majority of them. The key methodological innovation lies in modeling the meta-flip graph as the search space for tensor decompositions and leveraging its topological properties to guide efficient derivation of rank bounds. This approach yields state-of-the-art upper bounds for numerous formats and provides a new theoretical tool and perspective for advancing the long-standing quest to lower the asymptotic exponent ω of matrix multiplication. The results offer significant theoretical support for the design of faster matrix multiplication algorithms.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Search and Optimization: Metareasoning and Metaheuristics
• 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

Continuing recent investigations of bounding the tensor rank of matrix multiplication using flip graphs, we present here improved rank bounds for about thirty matrix formats.

Problem

• Improving tensor rank bounds for matrix multiplication
• Exploring meta flip graph for matrix formats
• Advancing investigations on flip graph applications

Innovation

• Meta Flip Graph exploration for matrix multiplication
• Improved tensor rank bounds via flip graphs
• Enhanced bounds for thirty matrix multiplication formats