Fast Matrix Multiplication via Ternary Meta Flip Graphs

TL;DR

This paper presents a GPU-accelerated method for discovering efficient matrix multiplication schemes with coefficients in {-1, 0, 1}, achieving new best ranks and expanding the understanding of ternary coefficient implementations.

Contribution

It introduces a novel meta flip graph algorithm that efficiently finds ternary matrix multiplication schemes, improving known ranks and providing open-source tools and results.

Findings

New best ranks for specific matrix formats.

Discovered 32 schemes matching known optimal ranks.

Achieved 30 rank improvements in the binary field.

Abstract

Matrix multiplication optimization remains a fundamental challenge in computational mathematics. This work introduces a novel approach that discovers matrix multiplication schemes whose coefficients are restricted to the set $\{-1, 0, 1\}$ (denoted $Z_T$), minimizing naive additive complexity for efficient hardware implementation. The core of the method is a GPU-accelerated meta flip graph algorithm that maintains ternary safety through specialized arithmetic operations and sign symmetry breaking. Key results include new best ranks for the formats $4 \times 5 \times 12$, $5 \times 6 \times 10$, and $6 \times 7 \times 9$, the independent discovery of 32 schemes in $Z_T$ that match known optimal ranks (including 8 previously known only with rational coefficients), and 30 rank improvements in the binary field. The analysis of 164 known schemes shows that 92 admit a ternary-coefficient implementation, while 72 could not be found under this constraint, defining the current boundaries of the approach. All software, results, and discovered schemes are provided as open-source.