Point source localisation with unbalanced optimal transport

TL;DR

This paper introduces a novel optimization method for point source localization that leverages unbalanced optimal transport in measure spaces, avoiding direct computation of transport distances and demonstrating improved numerical performance.

Contribution

It develops a new forward-backward optimization algorithm based on unbalanced optimal transport, avoiding explicit transport distance calculations and improving performance.

Findings

Algorithm shows improved numerical results

Uses transport three-plans to avoid direct distance computation

Derives from the geometry of measure spaces

Abstract

Replacing the quadratic proximal penalty familiar from Hilbert spaces by an unbalanced optimal transport distance, we develop forward-backward type optimisation methods in spaces of Radon measures. We avoid the actual computation of the optimal transport distances through the use of transport three-plans and the rough concept of transport subdifferentials. The resulting algorithm has a step similar to the sliding heuristics previously introduced for conditional gradient methods, however, now non-heuristically derived from the geometry of the space. We demonstrate the improved numerical performance of the approach.