Sparse super resolution is Lipschitz continuous

TL;DR

This paper proves that super resolution, modeled as a mapping from trigonometric moments to support and weights, is Lipschitz continuous, with explicit bounds depending on dimension and sampling effort, impacting stability analysis.

Contribution

It establishes the Lipschitz continuity of the super resolution map and provides explicit Lipschitz constant estimates, enhancing understanding of stability in super resolution problems.

Findings

Super resolution mapping is locally Lipschitz continuous.

Super resolution with Wasserstein distance is globally Lipschitz.

Improved estimate for smallest singular value of Vandermonde matrices.

Abstract

Motivated by the application of neural networks in super resolution microscopy, this paper considers super resolution as the mapping of trigonometric moments of a discrete measure on $[0,1)^d$ to its support and weights. We prove that this map satisfies a local Lipschitz property where we give explicit estimates for the Lipschitz constant depending on the dimension $d$ and the sampling effort. Moreover, this local Lipschitz estimate allows to conclude that super resolution with the Wasserstein distance as the metric on the parameter space is even globally Lipschitz continuous. As a byproduct, we improve an estimate for the smallest singular value of multivariate Vandermonde matrices having pairwise clustering nodes.