# The dual approach to non-negative super-resolution: impact on primal reconstruction accuracy

## TL;DR

This paper investigates the stability of non-negative super-resolution solutions by analyzing the dual approach, establishing a relationship between perturbations in dual and primal variables using advanced mathematical techniques.

## Contribution

It provides a novel stability analysis of the dual approach in non-negative super-resolution, linking dual perturbations to primal solution accuracy.

## Key findings

- Established a quantitative relationship between dual and primal perturbations.
- Demonstrated stability of super-resolution solutions under dual problem perturbations.
- Applied a non-trivial implicit function theorem to analyze solution stability.

## 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 recently shown that exact recovery is possible by minimising the total variation norm of the measure. An alternative practical approach is to solve its dual. In this paper, we study the stability of solutions with respect to the solutions to the dual problem. In particular, we establish a relationship between perturbations in the dual variable and the primal variables around the optimiser. This is achieved by applying a quantitative version of the implicit function theorem in a non-trivial way.