PittsburghV1, Main, Exploration, bibRecord, 000016

Estimating the Reach of a Manifold

Identifieur interne : 000016 ( Main/Exploration ); précédent : 000015; suivant : 000017

Estimating the Reach of a Manifold

Auteurs : Eddie Aamari [France] ; Jisu Kim [États-Unis] ; Frédéric Chazal [France] ; Bertrand Michel [France] ; Alessandro Rinaldo [États-Unis] ; Larry Wasserman [États-Unis]

Source :

RBID : Hal:hal-01521955

English descriptors

mix :
- Curvature, Geometric Inference, Minimax Estimation, Reach.

Abstract

Various problems in manifold estimation make use of a quantity called the {\em reach}, denoted by $\tau_M$, which is a measure of the regularity of the manifold. This paper is the first investigation into the problem of how to estimate the reach. First, we study the geometry of the reach through an approximation perspective. We derive new geometric results on the reach for submanifolds without boundary. An estimator $\hat{\tau}$ of $\tau_M$ is proposed in a framework where tangent spaces are known, and bounds assessing its efficiency are derived. In the case of i.i.d. random point cloud $\mathbb{X}_n$, $\hat{\tau}(\mathbb{X}_n)$ is showed to achieve uniform expected loss bounds over a $\mathcal{C}^3$-like model. Finally, we obtain upper and lower bounds on the minimax rate for estimating the reach.

Url:

https://hal.archives-ouvertes.fr/hal-01521955

Affiliations:

France, États-Unis

Links toward previous steps (curation, corpus...)

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to stream Hal, to step Curation: 000256
to stream Hal, to step Checkpoint: 000016
to stream Main, to step Merge: 000016
to stream Main, to step Curation: 000016

Le document en format XML

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<front><div type="abstract" xml:lang="en">Various problems in manifold estimation make use of a quantity called the {\em reach}, denoted by $\tau_M$, which is a measure of the regularity of the manifold. This paper is the first investigation into the problem of how to estimate the reach. First, we study the geometry of the reach through an approximation perspective. We derive new geometric results on the reach for submanifolds without boundary. An estimator $\hat{\tau}$ of $\tau_M$ is proposed in a framework where tangent spaces are known, and bounds assessing its efficiency are derived. In the case of i.i.d. random point cloud $\mathbb{X}_n$, $\hat{\tau}(\mathbb{X}_n)$ is showed to achieve uniform expected loss bounds over a $\mathcal{C}^3$-like model. Finally, we obtain upper and lower bounds on the minimax rate for estimating the reach.</div>
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Estimating the Reach of a Manifold

Estimating the Reach of a Manifold

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Abstract

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