On the singular values of matrices with displacement structure

TL;DR

This paper derives explicit bounds on the singular values of matrices with displacement structure, showing how they can be approximated by low-rank matrices, which has implications for numerical analysis and applications.

Contribution

It introduces a new extremal problem approach to bound singular values of structured matrices, providing explicit decay rates and low-rank approximation results.

Findings

Bound on the kth singular value of positive definite Hankel matrices

Low-rank approximation of structured matrices with logarithmic rank growth

Extension of results to various classes of structured matrices

Abstract

Matrices with displacement structure such as Pick, Vandermonde, and Hankel matrices appear in a diverse range of applications. In this paper, we use an extremal problem involving rational functions to derive explicit bounds on the singular values of such matrices. For example, we show that the $k$th singular value of a real $n\times n$ positive definite Hankel matrix, $H_n$, is bounded by $C\rho^{-k/\log n}\|H\|_2$ with explicitly given constants $C>0$ and $\rho>1$, where $\|H_n\|_2$ is the spectral norm. This means that a real $n\times n$ positive definite Hankel matrix can be approximated, up to an accuracy of $\epsilon\|H_n\|_2$ with $0<\epsilon<1$, by a rank $\mathcal{O}(\log n\log(1/\epsilon) )$ matrix. Analogous results are obtained for Pick, Cauchy, real Vandermonde, L\"{o}wner, and certain Krylov matrices.