The Annals of Statistics

Nonasymptotic rates for manifold, tangent space and curvature estimation

Eddie Aamari and Clément Levrard

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Abstract

Given a noisy sample from a submanifold $M\subset\mathbb{R}^{D}$, we derive optimal rates for the estimation of tangent spaces $T_{X}M$, the second fundamental form $\mathit{II}_{X}^{M}$ 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.

Article information

Source
Ann. Statist., Volume 47, Number 1 (2019), 177-204.

Dates
Received: April 2017
Revised: October 2017
First available in Project Euclid: 30 November 2018

Permanent link to this document
https://projecteuclid.org/euclid.aos/1543568586

Digital Object Identifier
doi:10.1214/18-AOS1685

Mathematical Reviews number (MathSciNet)
MR3909931

Zentralblatt MATH identifier
07036199

Subjects
Primary: 62G05: Estimation 62C20: Minimax procedures

Keywords
Geometric inference minimax manifold learning

Citation

Aamari, Eddie; Levrard, Clément. Nonasymptotic rates for manifold, tangent space and curvature estimation. Ann. Statist. 47 (2019), no. 1, 177--204. doi:10.1214/18-AOS1685. https://projecteuclid.org/euclid.aos/1543568586


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Supplemental materials

  • Appendix: Geometric background and proofs of intermediate results. Due to space constraints, we relegate technical details of the remaining proofs to the supplement [2].