Unbalanced Optimal Transport: Dynamic and Kantorovich Formulation
Lenaic Chizat, Gabriel Peyré, Bernhard Schmitzer, François-Xavier Vialard
TL;DR
The article develops a unified framework for unbalanced optimal transport by pairing dynamic formulations, which incorporate mass creation and destruction via a continuity equation with source, with static Kantorovich-type formulations using semi-couplings. It defines a broad class of dynamic costs through a convex, 1-homogeneous infinitesimal cost f and proves a duality and metric structure, including existence and geodesic properties. A central result is the equivalence between the dynamic distance $C_D$ and a convexified static distance $C_K$ via the minimal path cost, bridging the two viewpoints similarly to Benamou–Brenier, and enabling efficient static optimization schemes. The paper applies the theory to concrete cases, notably Optimal Partial Transport and the Wasserstein-Fisher-Rao metric, and establishes important limits and Γ-convergence results linking WFR to classical OT, with implications for numerical methods and analysis of unbalanced transport problems.
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.