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Mapping Estimation for Discrete Optimal Transport
Michaël Perrot · Nicolas Courty · Rémi Flamary · Amaury Habrard

Tue Dec 06 09:00 AM -- 12:30 PM (PST) @ Area 5+6+7+8 #137
We are interested in the computation of the transport map of an Optimal Transport problem. Most of the computational approaches of Optimal Transport use the Kantorovich relaxation of the problem to learn a probabilistic coupling $\mgamma$ but do not address the problem of learning the underlying transport map $\funcT$ linked to the original Monge problem. Consequently, it lowers the potential usage of such methods in contexts where out-of-samples computations are mandatory. In this paper we propose a new way to jointly learn the coupling and an approximation of the transport map. We use a jointly convex formulation which can be efficiently optimized. Additionally, jointly learning the coupling and the transport map allows to smooth the result of the Optimal Transport and generalize it to out-of-samples examples. Empirically, we show the interest and the relevance of our method in two tasks: domain adaptation and image editing.

Author Information

Michaël Perrot (University of Saint-Etienne)
Nicolas Courty (IRISA / University South Brittany)
Rémi Flamary (École Polytechnique)
Amaury Habrard (University of Saint-Etienne)

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