Regularity theory and geometry of unbalanced optimal transport

Using the dual formulation only, we show that regularity of unbalanced optimal transport also called entropy-transport inherits from regularity of standard optimal transport. We then provide detailed examples of Riemannian manifolds and costs for which unbalanced optimal transport is regular.Among a...

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Bibliographic Details
Published in:arXiv.org 2021-12
Main Authors: Gallouët, Thomas, Ghezzi, Roberta, Vialard, François-Xavier
Format: Article
Language:English
Subjects:
Regularity
Riemann manifold
Online Access:Get full text
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Summary:Using the dual formulation only, we show that regularity of unbalanced optimal transport also called entropy-transport inherits from regularity of standard optimal transport. We then provide detailed examples of Riemannian manifolds and costs for which unbalanced optimal transport is regular.Among all entropy-transport formulations, Wasserstein-Fisher-Rao metric, also called Hellinger-Kantorovich, stands out since it admits a dynamic formulation, which extends the Benamou-Brenier formulation of optimal transport. After demonstrating the equivalence between dynamic and static formulations on a closed Riemannian manifold, we prove a polar factorization theorem, similar to the one due to Brenier and Mc-Cann. As a byproduct, we formulate the Monge-Amp{è}re equation associated with Wasserstein-Fisher-Rao metric, which also holds for more general costs.
ISSN:2331-8422