no code implementations • 19 May 2022 • François-Pierre Paty, Philippe Choné, Francis Kramarz
The theory of weak optimal transport (WOT), introduced by [Gozlan et al., 2017], generalizes the classic Monge-Kantorovich framework by allowing the transport cost between one point and the points it is matched with to be nonlinear.
no code implementations • ICML 2020 • François-Pierre Paty, Marco Cuturi
In this work we depart from this practical perspective and propose a new interpretation of regularization as a robust mechanism, and show using Fenchel duality that any convex regularization of OT can be interpreted as ground cost adversarial.
no code implementations • 26 May 2019 • François-Pierre Paty, Alexandre d'Aspremont, Marco Cuturi
On the other hand, one of the greatest achievements of the OT literature in recent years lies in regularity theory: Caffarelli showed that the OT map between two well behaved measures is Lipschitz, or equivalently when considering 2-Wasserstein distances, that Brenier convex potentials (whose gradient yields an optimal map) are smooth.
no code implementations • 25 Jan 2019 • François-Pierre Paty, Marco Cuturi
Making sense of Wasserstein distances between discrete measures in high-dimensional settings remains a challenge.