Electronic Journal of Statistics

Estimating the reach of a manifold

Eddie Aamari, Jisu Kim, Frédéric Chazal, Bertrand Michel, Alessandro Rinaldo, and Larry Wasserman

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Abstract

Various problems in manifold estimation make use of a quantity called the 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 an oracle 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.

Article information

Source
Electron. J. Statist., Volume 13, Number 1 (2019), 1359-1399.

Dates
Received: March 2018
First available in Project Euclid: 12 April 2019

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1555056153

Digital Object Identifier
doi:10.1214/19-EJS1551

Subjects
Primary: 62G05: Estimation
Secondary: 62C20: Minimax procedures 68U05: Computer graphics; computational geometry [See also 65D18]

Keywords
Geometric inference reach minimax risk

Rights
Creative Commons Attribution 4.0 International License.

Citation

Aamari, Eddie; Kim, Jisu; Chazal, Frédéric; Michel, Bertrand; Rinaldo, Alessandro; Wasserman, Larry. Estimating the reach of a manifold. Electron. J. Statist. 13 (2019), no. 1, 1359--1399. doi:10.1214/19-EJS1551. https://projecteuclid.org/euclid.ejs/1555056153


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