A characterization of the Non-Degenerate Source Condition in Super-Resolution
This work provides theoretical guarantees for support stability in super-resolution, benefiting researchers in signal processing and inverse problems.
The paper characterizes the Non-Degenerate Source Condition (NDSC) for super-resolution, providing necessary and sufficient conditions for support stability. It shows that for the Laplace kernel, NDSC holds unconditionally with at least 2M measurements, and for the Gaussian filter, it holds under two distinct sampling configurations.
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{é} 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.