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

Eddie Aamari; Clément Levrard

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arXiv · 1705.00989

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

Eddie Aamari·Clément Levrard

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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 H{\"o}lder 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.

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BibTeX RIS

Eddie Aamari, Clément Levrard. 2017-05-02. Non-Asymptotic Rates for Manifold, Tangent Space, and Curvature Estimation. https://arxiv.org/abs/1705.00989

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