On Adaptive Confidence Sets for the Wasserstein Distances

Neil Deo (1) , Thibault Randrianarisoa (2)

1 DPMMS - Department of Pure Mathematics and Mathematical Statistics
2 LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation

In the density estimation model, we investigate the problem of constructing adaptive honest confidence sets with diameter measured in Wasserstein distance Wp, p >=1, and for densities with unknown regularity measured on a Besov scale. As sampling domains, we focus on the d-dimensional torus Td, in which case 1<=p<= 2, and Rd, for which p = 1. We identify necessary and sufficient conditions for the existence of adaptive confidence sets with diameters of the order of the regularity-dependent Wp-minimax estimation rate. Interestingly, it appears that the possibility of such adaptation of the diameter depends on the dimension of the underlying space. In low dimensions, d<= 4, adaptation to any regularity is possible. In higher dimensions, adaptation is possible if and only if the underlying regularities belong to some interval of width at least d/(d-4). This contrasts with the usual Lp-theory where, independently of the dimension, adaptation occurs only if regularities lie in a small fixed-width window. For configurations allowing these adaptive sets to exist, we explicitly construct confidence regions via the method of risk estimation. These are the first results in a statistical approach to adaptive uncertainty quantification with Wasserstein distances. Our analysis and methods extend to weak losses such as Sobolev norms with negative smoothness indices.

Keywords: Wasserstein distance; Uncertainty Quantification; Nonparametric confidence sets

Domains: Statistics [stat]; Mathematics [math]

Neil Deo, Thibault Randrianarisoa. On Adaptive Confidence Sets for the Wasserstein Distances. Bernoulli, 2023. ⟨hal-03890826⟩