Point source localisation with unbalanced optimal transport

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.