arXiv:2402.01011cs.CC
Ruling Out Low-rank Matrix Multiplication Tensor Decompositions with Symmetries via SAT
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Abstract
We analyze rank decompositions of the $3\times 3$ matrix multiplication tensor over $\mathbb{Z}/2\mathbb{Z}$. We restrict our attention to decompositions of rank $\le 21$, as only those decompositions will yield an asymptotically faster algorithm for matrix multiplication than Strassen's algorithm. To reduce search space, we also require decompositions to have certain symmetries. Using Boolean SAT solvers, we show that under certain symmetries, such decompositions do not exist.
pretty much all progress in theoretical matmul comes from approximate decomposition tools, so results happen only when error is small enough to get fixed easily - on large multiplications on the small side even optimal (3x3)(3x3) matmul is unknown more precisely than "between 19…
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