Statistique - Probabilités - Optimisation et Contrôle

Minimax adaptive estimation in manifold inference

by Mr Vincent Divol (INRIA)

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We focus on the problem of manifold estimation: given n i.i.d. observations X_1 , . . . , X_n in R^D sampled according to a law P supported close to some unknown manifold M ⊂ R^D , the goal of the statistician is to reconstruct the manifold using the observations. Minimax rates for this problem have been obtained under various model assumptions on the law P and the regularity of the underlying manifold M . The corresponding minimax estimators which have been proposed so far all depend crucially on the a priori knowledge of some parameters quantifying the regularity of M (such as its reach), whereas the statistician will not have access to those quantities in practice. Our contribution to the matter is twofold: first, we introduce a one-parameter family of estimators ( M̂ t ) t≥0 , and show that for some choice of t (depending on the regularity parameters), the corresponding estimator is minimax on the class of models of C^2 manifolds. Second, we propose a completely data-driven selection procedure for the parameter t, leading to an adaptive minimax estimator on this class of models.