A characterization of the Non-Degenerate Source Condition in super-resolution. (6th March 2019) - Duval, Vincent
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 that was introduced by Duval and Peyré to ensure support stability in super-resolution. We provide sufficient conditions that, 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 that approximate the uniform Lebesgue measure or, more surprisingly, samples that are all confined in a sufficiently small interval.