Improving the numerical stability of fast matrix multiplication

TL;DR

This paper enhances the numerical stability of fast matrix multiplication algorithms, making them more practical by improving accuracy through theoretical analysis, algorithmic modifications, and input scaling, supported by empirical benchmarks.

Contribution

It provides a generalized error analysis of fast algorithms, proposes techniques to improve their accuracy, and demonstrates practical improvements through benchmarking.

Findings

Improved error bounds for fast matrix multiplication algorithms.

Algorithmic modifications and input scaling enhance numerical stability.

Empirical benchmarks confirm increased accuracy in practical scenarios.

Abstract

Fast algorithms for matrix multiplication, namely those that perform asymptotically fewer scalar operations than the classical algorithm, have been considered primarily of theoretical interest. Apart from Strassen's original algorithm, few fast algorithms have been efficiently implemented or used in practical applications. However, there exist many practical alternatives to Strassen's algorithm with varying performance and numerical properties. Fast algorithms are known to be numerically stable, but because their error bounds are slightly weaker than the classical algorithm, they are not used even in cases where they provide a performance benefit. We argue in this paper that the numerical sacrifice of fast algorithms, particularly for the typical use cases of practical algorithms, is not prohibitive, and we explore ways to improve the accuracy both theoretically and empirically. The numerical accuracy of fast matrix multiplication depends on properties of the algorithm and of the input matrices, and we consider both contributions independently. We generalize and tighten previous error analyses of fast algorithms and compare their properties. We discuss algorithmic techniques for improving the error guarantees from two perspectives: manipulating the algorithms, and reducing input anomalies by various forms of diagonal scaling. Finally, we benchmark performance and demonstrate our improved numerical accuracy.