The Annals of Statistics

Nonasymptotic rates for manifold, tangent space and curvature estimation

Eddie Aamari and Clément Levrard

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G05: Estimation 62C20: Minimax procedures

Geometric inference minimax manifold learning


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.

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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].