文章基本信息

标题：Estimating the reach of a manifold
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作者：Eddie Aamari ; Jisu Kim ; Frédéric Chazal 等
期刊名称：Electronic Journal of Statistics
印刷版ISSN：1935-7524
出版年度：2019
卷号：13
期号：1
页码：1359-1399
DOI：10.1214/19-EJS1551
语种：English
出版社：Institute of Mathematical Statistics
摘要：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.