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.