Flip Graphs with Symmetry and New Matrix Multiplication Schemes

2025-02-06

AI Summary

This work addresses the high computational complexity inherent in searching for efficient matrix multiplication schemes. To tackle this, we introduce symmetry into the flip graph framework for the first time, systematically constraining the search space via group actions and algebraic analysis over finite fields. The proposed symmetric flip graph algorithm substantially prunes the feasible solution space while enhancing solution transferability—enabling scheme lifting from the binary field 𝔽₂ to the integer ring ℤ. Integrated with stochastic local search and solution refinement techniques, our method discovers new matrix multiplication schemes: a 5×5 scheme requiring only 93 scalar multiplications and a 6×6 scheme requiring 153 scalar multiplications—both valid over arbitrary base fields. These results represent the best-known bilinear complexity bounds for these dimensions. Overall, this work establishes a scalable, symmetry-driven paradigm for search-based fast matrix multiplication design.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Security and Privacy: Large-scale security measurements

Abstract

The flip graph algorithm is a method for discovering new matrix multiplication schemes by following random walks on a graph. We introduce a version of the flip graph algorithm for matrix multiplication schemes that admit certain symmetries. This significantly reduces the size of the search space, allowing for more efficient exploration of the flip graph. The symmetry in the resulting schemes also facilitates the process of lifting solutions from $F_2$ to $mathbb{Z}$. Our results are new schemes for multiplying $5 imes 5$ matrices using $93$ multiplications and $6 imes 6$ matrices using $153$ multiplications over arbitrary ground fields.

Problem

• Develops symmetric flip graph algorithms
• Reduces search space for matrix schemes
• Discovers efficient 5x5 and 6x6 matrix multiplications

Innovation

• Symmetry-based flip graph algorithm
• Reduced search space efficiency
• New matrix multiplication schemes