The dual approach to non-negative super-resolution: perturbation analysis

TL;DR

This paper analyzes the stability of non-negative super-resolution solutions, linking perturbations in dual variables to changes in primal solutions and measurement noise, using a quantitative implicit function theorem.

Contribution

It provides a novel stability analysis of super-resolution solutions with respect to dual problem perturbations and measurement noise.

Findings

Established a relationship between dual and primal perturbations.

Quantified the effect of measurement noise on solution stability.

Applied a quantitative implicit function theorem for analysis.

Abstract

We study the problem of super-resolution, where we recover the locations and weights of non-negative point sources from a few samples of their convolution with a Gaussian kernel. It has been shown that exact recovery is possible by minimising the total variation norm of the measure, and a practical way of achieve this is by solving the dual problem. In this paper, we study the stability of solutions with respect to the solutions dual problem, both in the case of exact measurements and in the case of measurements with additive noise. In particular, we establish a relationship between perturbations in the dual variable and perturbations in the primal variable around the optimiser and a similar relationship between perturbations in the dual variable around the optimiser and the magnitude of the additive noise in the measurements. Our analysis is based on a quantitative version of the implicit function theorem.