Nonasymptotic rates for manifold, tangent space and curvature estimation

By Eddie Aamari, Clément Levrard. Annals of Statistics, 2017

93 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/ad57226e6b7d308288ea0b131196bf1f2ec400b0

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. Alessandro Rinaldo
2. Bertrand Michel
3. Frédéric Chazal

In Computational Theory and Mathematics

1. Frédéric Chazal
2. Bertrand Michel
3. Alessandro Rinaldo
4. Eddie Aamari

In Geometry and Topology

1. Alessandro Rinaldo
2. Bertrand Michel
3. Frédéric Chazal

Citations in the corpus

• Estimating the reach of a manifold (2017, background): “r, Section6is only a rst step towards a framework where tangent spaces are unknown. A minimax upper bound in this case is still an open question. Considering smoother Ck models (k3) such as those of [2], or data with additive noise would also be of interest. In practice, since large reach ensures re”