Rates of convergence for robust geometric inference

TL;DR

This paper establishes precise convergence rates for the empirical distance to a measure (DTEM) used in topological data analysis, linking the rates to local geometric properties of the data support.

Contribution

It provides a tight deviation bound for DTEM, connecting convergence rates to the regularity of a key quantile function reflecting local geometry.

Findings

Convergence rates depend on the regularity of a specific quantile function.

Numerical experiments confirm the tightness of the theoretical bounds.

The analysis offers insights into the difficulty of geometric inference problems.

Abstract

Distances to compact sets are widely used in the field of Topological Data Analysis for inferring geometric and topological features from point clouds. In this context, the distance to a probability measure (DTM) has been introduced by Chazal et al. (2011) as a robust alternative to the distance a compact set. In practice, the DTM can be estimated by its empirical counterpart, that is the distance to the empirical measure (DTEM). In this paper we give a tight control of the deviation of the DTEM. Our analysis relies on a local analysis of empirical processes. In particular, we show that the rates of convergence of the DTEM directly depends on the regularity at zero of a particular quantile fonction which contains some local information about the geometry of the support. This quantile function is the relevant quantity to describe precisely how difficult is a geometric inference problem. Several numerical experiments illustrate the convergence of the DTEM and also confirm that our bounds are tight.