Butterfly factorization with error guarantees

TL;DR

This paper introduces a new butterfly factorization method with error guarantees, providing a formal framework, identifying optimal supports, and offering an algorithm with strong theoretical bounds on approximation error.

Contribution

It proposes a novel mathematical description of butterfly factors, introduces the concept of chainability for optimal supports, and develops an algorithm with provable error bounds.

Findings

The new algorithm achieves bounded approximation error independent of the target matrix.

Identification of supports that guarantee the existence of an optimal butterfly factorization.

Theoretical guarantees surpass existing methods in approximation quality.

Abstract

In this paper, we investigate the butterfly factorization problem, i.e., the problem of approximating a matrix by a product of sparse and structured factors. We propose a new formal mathematical description of such factors, that encompasses many different variations of butterfly factorization with different choices of the prescribed sparsity patterns. Among these supports, we identify those that ensure that the factorization problem admits an optimum, thanks to a new property called ``chainability''. For those supports we propose a new butterfly algorithm that yields an approximate solution to the butterfly factorization problem and that is supported by stronger theoretical guarantees than existing factorization methods. Specifically, we show that the ratio of the approximation error by the minimum value is bounded by a constant, independent of the target matrix.