🤖 AI Summary
This work addresses the absence of a unified, verifiable catalog for small-scale fast matrix multiplication algorithms scattered across diverse domains with inconsistent formats and naming conventions. The authors construct a comprehensive algorithm repository covering all instances up to size 32×32×32, supporting multiple number fields and commutative variants. By introducing the notion of “non-overlappingness,” they clearly distinguish between discovering novel bilinear kernels and composing existing ones, thereby decoupling algorithm discovery from composition and resolving attribution disputes in the literature. Leveraging a state-of-the-art closure-search framework augmented with techniques such as axis flipping, Kronecker products, axis concatenation, random products, distributive recomposition (including output stripping and pair fusion), and downward projection, the study systematically recombines and extends known algorithms, yielding numerous new low-rank schemes—including ternary integer algorithms—and automatically generates DIS09 comparison tables categorized by number field and commutativity.
📝 Abstract
The 2022--2026 burst of activity in small-format matrix multiplication (AlphaTensor 2022, AlphaEvolve 2025, Schwartz--Zwecher 2025) has produced striking individual results but scattered them across different fields, attribution conventions, and serialisation formats. A complementary line of work -- Perminov's open-source flip-graph framework~\cite{perminov2026fast,perminov2025fast} -- instead drives existing construction methods, notably flip-graph and \emph{meta-flip-graph} search, at scale across large format spaces, discovering many new low-rank schemes (including ternary-integer ones) that further enrich the landscape this catalog must unify. We present a unified, machine-checkable catalog covering shapes up to \nmpshape{32}{32}{32} over \Rationals, \Integers, \Reals, \Complex, and \Ftwo, with a separate axis for commutative algorithms (Waksman 1970, Makarov 1986, Rosowski 2019).
Derivation over this catalog is performed by a \emph{frontier-closure search} that recombines catalog entries by axis-flip, Kronecker, axis concatenation, serendipitous products, recombination-with-allocation (with optional output peeling and pair fusion), and downward projection.
A central methodological point is the \emph{non-overlap property}: our recombination does not, and cannot, rediscover the shared bilinear products that hand-crafted constructions (Strassen, Laderman, Smirnov, AlphaTensor) are built around. This draws a clean line between the ``find a cleverer bilinear core'' and ``compose known cores'' axes of progress, and resolves several attribution puzzles in the literature.
We refresh the DIS09 comparison tables, split per field and with a commutative column, and provide the tooling to regenerate them automatically as the catalog evolves.