Appears in collection : Séminaire Parisien de Statistique

Assume that we observe i.i.d. points lying close to some unknown d-dimensional submanifold of class C^k in a possibly high-dimensional space R^D. We study the problem of reconstructing the probability distribution generating the sample. After remarking that this problem is degenerate for a large class of standard losses (L_p, Hellinger, Kulback-Leibler, etc.), we focus on the Wassterstein loss, for which we build an estimator, based on kernel density estimation, whose rate of convergence depends both on d, k and on the regularity s of the underlying density, but not on the ambient dimension D. This estimator is shown to be minimax and attains considerably faster rate than the naive empirical estimator.