A characterization of the Non-Degenerate Source Condition in Super-Resolution

TL;DR

This paper characterizes the Non-Degenerate Source Condition in super-resolution, providing necessary and sufficient conditions for support stability, with implications for Gaussian and Laplace kernels under various sampling schemes.

Contribution

It establishes a precise criterion for the NDSC, unifying previous sufficient conditions and extending understanding to new sampling configurations.

Findings

Conditions hold unconditionally for Laplace kernel with at least 2M measurements.

NDSC fulfilled for Gaussian filter with uniform or small interval sampling.

Provides a necessary and sufficient condition for support stability in super-resolution.

Abstract

In a recent article, Schiebinger et al. provided sufficient conditions for the noiseless recovery of a signal made of M Dirac masses given 2M + 1 observations of, e.g. , its convolution with a Gaussian filter, using the Basis Pursuit for measures. In the present work, we show that a variant of their criterion provides a necessary and sufficient condition for the Non-Degenerate Source Condition (NDSC) which was introduced by Duval and Peyr{\'e} to ensure support stability in super-resolution. We provide sufficient conditions which, for instance, hold unconditionally for the Laplace kernel provided one has at least 2M measurements. For the Gaussian filter, we show that those conditions are fulfilled in two very different configurations: samples which approximate the uniform Lebesgue measure or, more surprisingly, samples which are all confined in a sufficiently small interval.