A characterization of the Non-Degenerate Source Condition in super-resolution. (6th March 2019)

• Record Type: Journal Article
• Authors: Duval, Vincent
• Is Part Of: Information and inference. Volume 9:Number 1(2020)
• Page Start: 235
• Page End: 269
• Publication Date: 2019-03-06
• Subjects: super-resolution -- ℓ1 norm -- total variation minimization -- LASSO for measures -- T-systems; Mathematical models -- Periodicals; 519.605
• DOI: 10.1093/imaiai/iaz002
• Languages: English
• ISSNs: 2049-8764

Abstract

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.