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.

翻译：许多与快速变换相关的矩阵具有某种低秩特性，其特征为矩阵存在多个分块划分，且每个分块均为低秩。在已知这些分块划分的前提下，存在一类称为蝴蝶因式分解的算法，可将矩阵近似为稀疏因子的乘积，从而实现关联线性算子的快速求值。本文提出一种新方法，能够在不依赖矩阵元素解析假设的条件下，通过代数手段识别可进行蝴蝶因式分解的矩阵的分块划分。