Non-Asymptotic Rates for Manifold, Tangent Space, and Curvature Estimation

Eddie Aamari 1, 2, 3 Clément Levrard 4
1 DATASHAPE - Understanding the Shape of Data
CRISAM - Inria Sophia Antipolis - Méditerranée , Inria Saclay - Ile de France
2 SELECT - Model selection in statistical learning
Inria Saclay - Ile de France, LMO - Laboratoire de Mathématiques d'Orsay, CNRS - Centre National de la Recherche Scientifique : UMR
Abstract : Given an $n$-sample drawn on a submanifold $M \subset \mathbb{R}^D$, we derive optimal rates for the estimation of tangent spaces $T_X M$, the second fundamental form $II_X^M$, and the submanifold $M$. After motivating their study, we introduce a quantitative class of $\mathcal{C}^k$-submanifolds in analogy with Hölder classes. The proposed estimators are based on local polynomials and allow to deal simultaneously with the three problems at stake. Minimax lower bounds are derived using a conditional version of Assouad's lemma when the base point $X$ is random.
Type de document :
Pré-publication, Document de travail
2017
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https://hal.archives-ouvertes.fr/hal-01516032
Contributeur : Clément Levrard <>
Soumis le : mardi 2 mai 2017 - 14:28:28
Dernière modification le : jeudi 20 juillet 2017 - 09:27:33

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  • HAL Id : hal-01516032, version 1
  • ARXIV : 1705.00989

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Eddie Aamari, Clément Levrard. Non-Asymptotic Rates for Manifold, Tangent Space, and Curvature Estimation. 2017. <hal-01516032>

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