Proximal methods for point source localisation

TL;DR

This paper extends proximal algorithms to measure spaces for point source localisation, providing convergence proofs and demonstrating numerical effectiveness beyond traditional Hilbert space methods.

Contribution

It introduces proximal-type methods like forward-backward splitting and primal-dual splitting for measure spaces, with rigorous convergence analysis.

Findings

Proximal methods outperform existing algorithms in numerical experiments.

Convergence proofs are established for measure space algorithms.

Methods are effective for practical point source localisation tasks.

Abstract

Point source localisation is generally modelled as a Lasso-type problem on measures. However, optimisation methods in non-Hilbert spaces, such as the space of Radon measures, are much less developed than in Hilbert spaces. Most numerical algorithms for point source localisation are based on the Frank-Wolfe conditional gradient method, for which ad hoc convergence theory is developed. We develop extensions of proximal-type methods to spaces of measures. This includes forward-backward splitting, its inertial version, and primal-dual proximal splitting. Their convergence proofs follow standard patterns. We demonstrate their numerical efficacy.