On the Uniqueness of Kantorovich Potentials
2022-01-20 | preprint
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- Authors
- Staudt, Thomas; Hundrieser, Shayan; Munk, Axel
- Abstract
- Kantorovich potentials denote the dual solutions of the renowned optimal transportation problem. Uniqueness of these solutions is relevant from both a theoretical and an algorithmic point of view, and has recently emerged as a necessary condition for asymptotic results in the context of statistical and entropic optimal transport. In this work, we challenge the common perception that uniqueness in continuous settings is reliant on the connectedness of the support of at least one of the involved measures, and we provide mild sufficient conditions for uniqueness even when both measures have disconnected support. Since our main finding builds upon the uniqueness of Kantorovich potentials on connected components, we revisit the corresponding arguments and provide generalizations of well-known results. Several auxiliary findings regarding the continuity of Kantorovich potentials, for example in geodesic spaces, are established along the way.
- Issue Date
- 20-January-2022
- Project
- EXC 2067: Multiscale Bioimaging
SFB 1456: Mathematik des Experiments: Die Herausforderung indirekter Messungen in den Naturwissenschaften
SFB 1456 | Cluster A | A04: Dynamics of cytoskeletal networks: From geometric structure to cell mechanics
SFB 1456 | Cluster C | C06: Optimal transport based colocalization - Working Group
- RG Munk