Practical fast matrix multiplication algorithms

Abstract

Matrix multiplication is a core building block for numerous scientific computing and, more recently, machine learning applications. Strassen's algorithm, the original Fast Matrix Multiplication (FMM) algorithm, has long fascinated computer scientists due to its startling property of reducing the number of computations required for multiplying n x n matrices from O(n³) to O(n [superscript 2.807]). Over the last half century, this has fueled many theoretical improvements such as other variations of Strassen-like FMM algorithms. Previous implementations of these FMM algorithms led to the "street wisdom" that they are only practical for large, relatively square matrices, that they require considerable workspace, and that they are difficult to achieve thread-level parallelism. The thesis of this work dispels these notions by demonstrating significant benefits for small and non-square matrices, requiring no workspace beyond what is already incorporated in high-performance implementations of matrix multiplication, and achieving performance benefits on multi-core, many-core, and distributed memory architectures.