Butterfly factorization by algorithmic identification of rank-one blocks

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.