Interpolative Decomposition Butterfly Factorization

TL;DR

This paper presents a kernel-independent interpolative decomposition butterfly factorization (IDBF) that provides a fast, data-sparse approximation for certain low-rank matrices, enabling rapid matrix-vector multiplication with broad applications.

Contribution

It introduces a novel IDBF framework that constructs a hierarchical, sparse matrix factorization in $O(N ext{log}N)$ time, applicable to a wide range of problems.

Findings

Constructs in $O(N ext{log}N)$ operations

Enables rapid $O(N ext{log}N)$ matrix-vector multiplication

Demonstrates effectiveness through numerical experiments

Abstract

This paper introduces a "kernel-independent" interpolative decomposition butterfly factorization (IDBF) as a data-sparse approximation for matrices that satisfy a complementary low-rank property. The IDBF can be constructed in $O(N\log N)$ operations for an $N\times N$ matrix via hierarchical interpolative decompositions (IDs), if matrix entries can be sampled individually and each sample takes $O(1)$ operations. The resulting factorization is a product of $O(\log N)$ sparse matrices, each with $O(N)$ non-zero entries. Hence, it can be applied to a vector rapidly in $O(N\log N)$ operations. IDBF is a general framework for nearly optimal fast matvec useful in a wide range of applications, e.g., special function transformation, Fourier integral operators, high-frequency wave computation. Numerical results are provided to demonstrate the effectiveness of the butterfly factorization and its construction algorithms.