Estimating the reach of a manifold via its convexity defect function
Abstract
The reach of a submanifold is a crucial regularity parameter for manifold learning and geometric inference from point clouds. This paper relates the reach of a submanifold to its convexity defect function. Using the stability properties of convexity defect functions, along with some new bounds and the recent submanifold estimator of Aamari and Levrard [Ann. Statist. 47 177-204 (2019)], an estimator for the reach is given. A uniform expected loss bound over a C^k model is found. Lower bounds for the minimax rate for estimating the reach over these models are also provided. The estimator almost achieves these rates in the C^3 and C^4 cases, with a gap given by a logarithmic factor.
Cite
@article{arxiv.2001.08006,
title = {Estimating the reach of a manifold via its convexity defect function},
author = {Clément Berenfeld and John Harvey and Marc Hoffmann and Krishnan Shankar},
journal= {arXiv preprint arXiv:2001.08006},
year = {2022}
}
Comments
35 pages, 4 figures. Various minor changes in v2 to correct minor errors and/or improve clarity. Thanks to excellent work by peer reviewers, in v3 an error in Lemma 4.9 was rectified, Section 4.2 was substantially revised and other minor changes made throughout and the manuscript was accepted for publication by Discrete & Computational Geometry. Extremely minor changes in v4