Un Mokameeting aura lieu le mercredi 12 janvier 2022 à 15h00, sur Discord.
Nous aurons le plaisir d’écouter deux exposés, l’un de Jonathan Niles-Weed (New York University), l’autre de Victor-Emmanuel Brunel (ENSAE/CREST).
Exposé de Jonathan Niles-Weed
Titre : Towards practical estimation of Brenier maps
Résumé : Given two probability distributions in R^d, a transport map is a function which maps samples from one distribution into samples from the other. For absolutely continuous measures, Brenier proved a remarkable theorem identifying a unique canonical transport map, which is “monotone” in a suitable sense. We study the question of whether this map can be efficiently estimated from samples. The minimax rates for this problem were recently established by Hutter and Rigollet (2021), but the estimator they propose is computationally infeasible in dimensions greater than three. We propose two new estimators—one minimax optimal, one not—which are significantly more practical to compute and implement. The analysis of these estimators is based on new stability results for the optimal transport problem and its regularized variants.
Exposé de Victor-Emmanuel Brunel
Titre : Statistical guarantees for high dimensional generative models
Résumé : We introduce a convenient framework for studying (adversarial) generative models from a statistical perspective. It consists in modeling the generative device as a smooth transformation of the unit hypercube of a dimension that is much smaller than that of the ambient space and measuring the quality of the generative model by means of an integral probability metric. In the particular case of an integral probability metric defined through a smoothness class, a risk bound is established, quantifying the role of various parameters. In particular, it clearly shows the impact of dimension reduction on the error of the generative model.
