On spectral and numerical properties of random butterfly matrices

TL;DR

This paper investigates the spectral and numerical characteristics of random butterfly matrices, highlighting their eigenvalue distribution and computational efficiency, with implications for randomized linear algebra.

Contribution

It provides a detailed analysis of the spectral properties of random butterfly matrices and explores their applications in efficient randomized linear algebra algorithms.

Findings

Eigenvalue distributions are uniform for certain classes.

Matrix-vector multiplication can be performed in O(N log N) time.

Connections to Haar measure on subgroups of the orthogonal group.

Abstract

Spectral and numerical properties of classes of random orthogonal butterfly matrices, as introduced by Parker (1995), are discussed, including the uniformity of eigenvalue distributions. These matrices are important because the matrix-vector product with an $N$-dimensional vector can be performed in $O(N \log N)$ operations. And in the simplest situation, these random matrices coincide with Haar measure on a subgroup of the orthogonal group. We discuss other implications in the context of randomized linear algebra.