Estimating the reach of a manifold
By Eddie Aamari, Jisu Kim, Frédéric Chazal, Bertrand Michel, Alessandro Rinaldo, Larry Wasserman LM-Orsay, Select, Datashape, U. Diego, Ecn and 1 more. Electronic Journal of Statistics, 2017
123 citations recorded by the source corpus.
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 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.
Subfields: Applied Mathematics, Computational Theory and Mathematics, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/8ed3df1dae92bb6fa009d362d2e181a086fe7c63
How researchers see this paper
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In Applied Mathematics
In Computational Theory and Mathematics
- Frédéric Chazal
- Bertrand Michel
- Paul Breiding
- Bernd Sturmfels
- Eric Katz
- Martyn Marshall
- Michael Joswig
- Jenia Tevelev
In Statistics and Probability
Citations in the corpus
- Learning algebraic varieties from samples (2018, methodology): “0;7!dim( ;). Remark 3.2. The idea of using dimension diagrams is inspired by persistent homology. Our dimension diagrams and our persistent homology barcodes of Section 4 both use in the interval [0;1] for the horizontal axis. This uniform scale for all samples makes comparisons across dierent datas”
- An Introduction to Topological Data Analysis: Fundamental and Practical Aspects for Data Scientists (2017, background): “Its estimation from random samples has been studied by Aamari et al. (2019). An important property of a distance-like function φ is that the topology of their sublevel sets φ can only change when r crosses a 0-critical value. Lemma 1 (isotopy lemma grove (1993)).”