2026-07-20T07:33:48Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:25898 2026-06-23T13:00:26Z ddc:004 open_access

Estimation of Conformal Metrics Taupin, Jérôme Geometric inference metric estimation conformal metric geodesics sets of positive reach 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(n²) 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Jérôme Taupin 2026 Is Part Of LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.SoCG.2026.92 urn:nbn:de:0030-drops-258986 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.92 eng https://creativecommons.org/licenses/by/4.0/legalcode