Fast Matrix Multiplication via Ternary Meta Flip Graphs

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.