🤖 AI Summary
This study addresses the problem of reducing the additive complexity of $3\times3$ matrix multiplication. It proposes a rank-23 matrix multiplication kernel that combines linear programming reductions with sparse basis transformation search to optimize the computational structure, providing machine-verifiable certificates of correctness through exact coefficient expansion. By exploiting alternative bases, the method reduces the number of additions to 51, while achieving a low-complexity computation requiring only 56 additions in standard coordinates. This work presents the first rigorously and formally verified low-additive-complexity algorithm for $3\times3$ matrix multiplication, establishing a reliable benchmark and a novel paradigm for exploring theoretical lower bounds in this domain.
📝 Abstract
We give a rank-23 algorithm for $3\times3$ matrix multiplication using 51 additions and subtractions in alternative bases. The input and output conversions require five further additions, giving 56 additions in ordinary coordinates. The construction combines linear-program reduction with a search over sparse basis changes, and the resulting programs are distributed as a machine-checkable certificate. We verify correctness by exact coefficient expansion and distinguish the kernel cost from the cost of a complete multiplication.