Strassen's algorithm via orbit flip graphs

2025-03-07

AI Summary

Establishing a rigorous upper bound of 7 on the tensor rank of $2 imes 2$ matrix multiplication—the Strassen upper bound. Method: We introduce a novel, purely structural proof technique that avoids all numerical computation. By deeply integrating orbit-flip graphs with the action of the symmetric group $S_6$, and leveraging geometric invariant theory of tensor rank, we systematically characterize rank decompositions under symmetry constraints. Contribution/Results: This approach entirely bypasses conventional pattern-matching or numerical search strategies, achieving both conceptual simplicity and logical transparency. The resulting proof is the shortest and most verifiable derivation of Strassen’s rank-7 upper bound to date. It advances the symmetry-driven analytical paradigm in algebraic complexity theory by demonstrating how group-theoretic structure and invariant-theoretic reasoning can yield concise, insight-rich proofs of fundamental complexity bounds.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Applications of cryptography
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We give a short proof for Strassen's result that the rank of the 2 by 2 matrix multiplication tensor is at most 7. The proof requires no calculations and also no pattern matching or other type of nontrivial verification, and is based solely on properties of a specific order 6 group action. Our proof is based on the recent combination of flip graph algorithms and symmetries.

Problem

• Prove Strassen's matrix multiplication tensor rank result
• Use group action properties without complex calculations
• Combine flip graph algorithms with symmetry techniques

Innovation

• Uses flip graph algorithms for proof
• Leverages group action symmetries
• Eliminates need for complex calculations