On the Stability of Super-Resolution and a Beurling-Selberg Type Extremal Problem

TL;DR

This paper investigates the stability conditions of super-resolution, linking the Fisher information matrix's eigenvalues to a Beurling-Selberg type extremal problem, and identifies a separation threshold for stable spike recovery.

Contribution

It introduces a new stability notion for the Fisher information matrix in super-resolution and connects it to a generalized Beurling-Selberg extremal problem, establishing a separation threshold.

Findings

Eigenvalues of the FIM can be bounded independently of the number of moments above a certain separation threshold.

A novel connection between FIM stability and Beurling-Selberg extremal problems is established.

Identifies a regime where super-resolution stability is guaranteed regardless of the number of moments.

Abstract

Super-resolution estimation is the problem of recovering a stream of spikes (point sources) from the noisy observation of a few numbers of its first trigonometric moments. The performance of super-resolution is recognized to be intimately related to the separation between the spikes to recover. A novel notion of stability of the Fisher information matrix (FIM) of the super-resolution problem is introduced when the minimal eigenvalue of the FIM is not asymptotically vanishing. The regime where the minimal separation is inversely proportional to the number of acquired moments is considered. It is shown that there is a separation threshold above which the eigenvalues of the FIM can be bounded by a quantity that does not depend on the number of moments. The proof relies on characterizing the connection between the stability of the FIM and a generalization of the Beurling-Selberg box approximation problem.