Unbalanced optimal transport: Dynamic and Kantorovich formulations

By Lénaïc Chizat, G. Peyr'e, Bernhard Schmitzer, François-Xavier Vialard. Journal of Functional Analysis, 2015

313 citations recorded by the source corpus.

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 [7] , [15] . Defined initially through a dynamic formulation, it belongs to this class of metrics and hence automatically benefits from a static Kantorovich formulation.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/f124d97459e402031e6f3731a871cff53b6a8c32

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Citations in the corpus

Robust Optimal Transport with Applications in Generative Modeling and Domain Adaptation (2020, methodology): “[6, 13] have proposed one such relaxation using f -divergence on marginal distributions.”
TrajectoryNet: A Dynamic Optimal Transport Network for Modeling Cellular Dynamics (2020, background): “Another direction that has recently received increased attention is in unbalanced optimal transport where the goal is to relax the problem to add and remove mass (Benamou, 2003; Chizat et al., 2018; Liero et al., 2018; Schiebinger et al., 2019).”