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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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Geometric inference reach minimax risk

Creative Commons Attribution 4.0 International License.


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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