Displacement smoothness of entropic optimal transport

Guillaume Carlier (1, 2) , Lénaïc Chizat (3) , Maxime Laborde (4)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 MOKAPLAN - Méthodes numériques pour le problème de Monge-Kantorovich et Applications en sciences sociales
3 EPFL - Ecole Polytechnique Fédérale de Lausanne
4 LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions

The function that maps a family of probability measures to the solution of the dual entropic optimal transport problem is known as the Schrödinger map. We prove that when the cost function is $C^{k+1}$ with k in N* then this map is Lipschitz continuous from the $L^2$-Wasserstein space to the space of $C^k$ functions. Our result holds on compact domains and covers the multi-marginal case. As applications, we prove displacement smoothness of the entropic optimal transport cost and the well-posedness of certain Wasserstein gradient flows involving this functional, including the Sinkhorn divergence and a multi-species system.

Mots clés: Entropic Optimal Transport; Schrödinger map; Wasserstein gradient flows

Domaines: Optimisation et contrôle [math.OC]

Guillaume Carlier, Lénaïc Chizat, Maxime Laborde. Displacement smoothness of entropic optimal transport. 2024. ⟨hal-03793562v2⟩