Séances du groupe de travail

Le groupe de travail est ouvert à tous ceux qui ont un intérêt de près ou de loin pour le transport optimal et ses applications. Les séances sont en général composées d’un exposé court (le premier) et d’un exposé long (le second).

Séance à venir: lundi 27 juin 2022.

UPEM, LAMA, 4B107 (quatrième étage du bâtiment Copernic), 13h - 15h.

1. Exposé de François-Pierre Paty sur des approches numériques pour le transport optimal faible et ses applications: Abstract: 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. In the so-called barycentric version of WOT, the cost for transporting a point x only depends on x and on the barycenter of the points it is matched with. This aggregation property of WOT is appealing in machine learning, economics and finance. Yet algorithms to compute WOT have only been developed for the special case of quadratic barycentric WOT, or depend on neural networks with no guarantee on the computed value and matching. The main difficulty lies in the transportation constraints which are costly to project onto. In this paper, we propose to use mirror descent algorithms to solve the primal and dual versions of the WOT problem. We also apply our algorithms to the variant of WOT introduced by [Choné et al., 2022] where mass is distributed from one space to another through unnormalized kernels (WOTUK). We empirically compare the solutions of WOT and WOTUK with classical OT. We illustrate our numerical methods to the economic framework of [Choné and Kramarz, 2021], namely the matching between workers and firms on labor markets.

2. Exposé d’Adrien Vacher sur l’estimation statistique et computationnelle des potentiels de transport en grande dimension à partir d’échantillons: Abstract: It was recently shown that under smoothness conditions, the squared Wasserstein distance between two distributions could be efficiently computed with appealing statistical error upper bounds. However, rather than the distance itself, the object of interest for applications such as generative modeling is the underlying optimal transport map. Hence, computational and statistical guarantees need to be obtained for the estimated maps themselves. In this paper, we propose the first tractable algorithm for which the statistical L2 error on the maps nearly matches the existing minimax lower-bounds for smooth map estimation. Our method is based on solving the semi-dual formulation of optimal transport with an infinite-dimensional sum-of-squares reformulation, and leads to an algorithm which has dimension-free polynomial rates in the number of samples, with potentially exponentially dimension-dependent constants.

Deuxième séance: lundi 23 mai 2022.

ENPC, salle de séminaire du CERMICS, 13h - 15h.

1. Exposé (seconde partie) de Paul-Marie Samson sur le problème de transport optimal faible (slides: Coûts de transport optimal faibles et dualité de Kantorovich).

2. Exposé de Benjamin Jourdain sur le transport optimal martingale et sa stabilité.

1ère séance: mardi 19 avril 2022.

Univ. Gustave Eiffel, Bâtiment Copernic, salle 2B140, 13h - 15h.

1. Exposé de Thomas Bonis sur la méthode de Stein Variational Gradient Descent (slides: Echantillonnage par descente variationnelle de Stein).

2. Exposé (première partie) de Paul-Marie Samson sur le problème de transport optimal faible (slides: Coûts de transport optimal faibles et dualité de Kantorovich).

François-Xavier Vialard

My research interests include optimal transport, large deformation by diffeomorphisms and their applications.