Minimax estimation of discontinuous optimal transport maps: The semi-discrete case - Archive ouverte HAL Accéder directement au contenu
Communication Dans Un Congrès Année : 2023

Minimax estimation of discontinuous optimal transport maps: The semi-discrete case

Résumé

We consider the problem of estimating the optimal transport map between two probability distributions, P and Q in R d , on the basis of i.i.d. samples. All existing statistical analyses of this problem require the assumption that the transport map is Lipschitz, a strong requirement that, in particular, excludes any examples where the transport map is discontinuous. As a first step towards developing estimation procedures for discontinuous maps, we consider the important special case where the data distribution Q is a discrete measure supported on a finite number of points in R d. We study a computationally efficient estimator initially proposed by Pooladian & Niles-Weed (2021), based on entropic optimal transport, and show in the semi-discrete setting that it converges at the minimax-optimal rate n-1/2 , independent of dimension. Other standard map estimation techniques both lack finite-sample guarantees in this setting and provably suffer from the curse of dimensionality. We confirm these results in numerical experiments, and provide experiments for other settings, not covered by our theory, which indicate that the entropic estimator is a promising methodology for other discontinuous transport map estimation problems.
Fichier principal
Vignette du fichier
pooladian23b.pdf (662.89 Ko) Télécharger le fichier
Origine : Fichiers produits par l'(les) auteur(s)

Dates et versions

hal-04352861 , version 1 (19-12-2023)

Identifiants

  • HAL Id : hal-04352861 , version 1

Citer

Aram-Alexandre Pooladian, Vincent Divol, Jonathan Niles-Weed. Minimax estimation of discontinuous optimal transport maps: The semi-discrete case. International Conference on Machine Learning, Jul 2023, Honolulu, United States. ⟨hal-04352861⟩
14 Consultations
9 Téléchargements

Partager

Gmail Facebook X LinkedIn More