Estimating the Reach of a Manifold via its Convexity Defect Function.

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. (1), 177-204 (2019)), an estimator for the reach is given. A uniform expected loss bound over a 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 and cases, with a gap given by a logarithmic factor.