Estimation of Conformal Metrics

TL;DR

This paper introduces a method for estimating conformal metrics on domains from point cloud data, demonstrating regularity, convergence rates, and graph equivalences under certain geometric and sampling assumptions.

Contribution

It provides a novel approach for estimating conformal metrics with provable convergence rates and links between graph constructions under regularity conditions.

Findings

Conformal geodesics exhibit good regularity properties.

Estimator achieves near-optimal convergence rates of n^(-1/d).

Equivalence between ball graphs and nearest-neighbors graphs under Ahlfors regularity.

Abstract

We study deformations of the geodesic distances on a domain of R N induced by a function called conformal factor. We show that under a positive reach assumption on the domain (not necessarily a submanifold) and mild assumptions on the conformal factor, geodesics for the conformal metric have good regularity properties in the form of a lower bounded reach. This regularity allows for efficient estimation of the conformal metric from a random point cloud with a relative error proportional to the Hausdorff distance between the point cloud and the original domain. We then establish convergence rates of order n^(-1/d) that are close to sharp when the intrinsic dimension d of the domain is large, for an estimator that can be computed in O(n^2 ) time. Finally, this paper includes a useful equivalence result between ball graphs and nearest-neighbors graphs when assuming Ahlfors regularity of the sampling measure, allowing to transpose results from one setting to another.