Brief Announcement: An Automatic Framework for High Performance Alternative Basis Fast Matrix Multiplication

Sub-cubic time matrix multiplication algorithms can be of practical use, provided that their arithmetic and communication costs are sufficiently small not only in the asymptotic sense, but already for useful input sizes. The alternative basis method has proved to be highly effective for reducing both costs in theory for all known feasible fast matrix multiplication algorithms. It has been tested and combined with other optimization techniques, such as the pebble game, only for Strassen's algorithm, demonstrating significant speedups in practice. The abundance of fast matrix multiplication algorithms and optimization techniques highlights the need for an automatic framework for accelerating the process of obtaining high performance code for them.We obtain an automatic framework for generating high-performance code for alternative basis fast matrix multiplication algorithms, utilizing various optimization techniques such as common subexpression elimination and a novel pebble game engine. These reduce the leading coefficients for arithmetic and communication costs by 20$ - 90%, and the memory footprint by approximately 50% - 70%. Using this framework, we optimize, implement, and benchmark a large set of algorithms, providing the first competitive high performance CPU implementations for many of them. The automatically generated code is typically faster than vendor-tuned GEMM, exceeding 50% in sequential configurations and 25% in parallel, and often outperforms the state-of-the-art code for Strassen's algorithm.