Interpolative Butterfly Factorization

TL;DR

This paper presents an interpolative butterfly factorization method that efficiently approximates certain harmonic analysis transforms with nearly optimal computational complexity, leveraging low-rank properties and structure-preserving compression.

Contribution

It introduces a novel sweeping matrix compression technique to produce an optimal butterfly factorization with $O(N ext{log}N)$ complexity, improving efficiency for harmonic analysis transforms.

Findings

Constructs an $O(N ext{log}N)$ complexity factorization

Uses structure-preserving low-rank approximations

Demonstrates effectiveness through numerical results

Abstract

This paper introduces the interpolative butterfly factorization for nearly optimal implementation of several transforms in harmonic analysis, when their explicit formulas satisfy certain analytic properties and the matrix representations of these transforms satisfy a complementary low-rank property. A preliminary interpolative butterfly factorization is constructed based on interpolative low-rank approximations of the complementary low-rank matrix. A novel sweeping matrix compression technique further compresses the preliminary interpolative butterfly factorization via a sequence of structure-preserving low-rank approximations. The sweeping procedure propagates the low-rank property among neighboring matrix factors to compress dense submatrices in the preliminary butterfly factorization to obtain an optimal one in the butterfly scheme. For an $N\times N$ matrix, it takes $O(N\log N)$ operations and complexity to construct the factorization as a product of $O(\log N)$ sparse matrices, each with $O(N)$ nonzero entries. Hence, it can be applied rapidly in $O(N\log N)$ operations. Numerical results are provided to demonstrate the effectiveness of this algorithm.