Unbalanced Optimal Transport: Dynamic and Kantorovich Formulation

TL;DR

This paper introduces a new class of distances between measures inspired by optimal transport, featuring dynamic and static formulations that enable flexible applications and include the Wasserstein-Fisher-Rao metric.

Contribution

It proposes a unified framework for unbalanced optimal transport distances with equivalent dynamic and static formulations, including the Wasserstein-Fisher-Rao metric.

Findings

Both formulations are convex optimization problems.

Switching between formulations benefits various applications.

Includes the Wasserstein-Fisher-Rao metric as a special case.

Abstract

This article presents a new class of distances between arbitrary nonnegative Radon measures inspired by optimal transport. These distances are defined by two equivalent alternative formulations: (i) a dynamic formulation defining the distance as a geodesic distance over the space of measures (ii) a static "Kantorovich" formulation where the distance is the minimum of an optimization problem over pairs of couplings describing the transfer (transport, creation and destruction) of mass between two measures. Both formulations are convex optimization problems, and the ability to switch from one to the other depending on the targeted application is a crucial property of our models. Of particular interest is the Wasserstein-Fisher-Rao metric recently introduced independently by Chizat et al. and Kondratyev et al. Defined initially through a dynamic formulation, it belongs to this class of metrics and hence automatically benefits from a static Kantorovich formulation.