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.
Chretien, S, A Thompson, and B Toader. 2019. “The Dual Approach to Non-Negative Super-Resolution: Impact on Primal Reconstruction Accuracy”, 13th International conference on Sampling Theory and Applications (SampTA) University of Bordeaux, 8-12 July 2019.
Acknowledgement of The Alan Turing Institute's contribution.