Universally consistent estimation of the reach

TL;DR

This paper introduces a universally consistent method to estimate the reach of a set in Euclidean space, with convergence rates under certain conditions, and discusses the limitations of finite sample detection of zero reach.

Contribution

It proposes the first universally consistent estimator of the reach, applicable with minimal assumptions, and analyzes its convergence and limitations.

Findings

The estimator is universally consistent with positive reach.

Convergence rates are established under additional assumptions.

It is impossible to detect zero reach from finite samples.

Abstract

The reach of a set $M \subset \mathbb R^d$, also known as condition number when $M$ is a manifold, was introduced by Federer in 1959. The reach is a central concept in geometric measure theory, set estimation, manifold learning, among others areas. We introduce a universally consistent estimate of the reach, just assuming that the reach is positive. Under an additional assumption we provide rates of convergence. We also show that it is not possible to determine, based on a finite sample, if the reach of the support of a density is zero or not. We provide a small simulation study and a bias correction method for the case when $M$ is a manifold.