Butterfly factorization by algorithmic identification of rank-one blocks

TL;DR

This paper introduces a novel algorithmic method to automatically identify low-rank block structures in matrices, facilitating efficient butterfly factorizations without prior analytical assumptions on the matrix entries.

Contribution

It presents a new algorithm to algebraically determine block partitionings for butterfly factorizations, expanding applicability to matrices with unknown low-rank structures.

Findings

Successfully identifies block partitionings in various matrices

Enables efficient butterfly factorization without prior structural knowledge

Improves computational speed for matrix-vector multiplications

Abstract

Many matrices associated with fast transforms posess a certain low-rank property characterized by the existence of several block partitionings of the matrix, where each block is of low rank. Provided that these partitionings are known, there exist algorithms, called butterfly factorization algorithms, that approximate the matrix into a product of sparse factors, thus enabling a rapid evaluation of the associated linear operator. This paper proposes a new method to identify algebraically these block partitionings for a matrix admitting a butterfly factorization, without any analytical assumption on its entries.