Document Open Access Logo

Estimation of Conformal Metrics

Author Jérôme Taupin

PDF

File

Thumbnail PDF
PDF
LIPIcs.SoCG.2026.92.pdf
  • Filesize: 0.72 MB
  • 15 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Geometric inference
  • metric estimation
  • conformal metric
  • geodesics
  • sets of positive reach

Metrics

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(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.

Cite As Get BibTex

Jérôme Taupin. Estimation of Conformal Metrics. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 92:1-92:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.SoCG.2026.92

Author Details

Jérôme Taupin
  • Université Paris-Saclay, France
  • INRIA Saclay, Palaiseau, France

References

  1. Eddie Aamari, Clément Berenfeld, and Clément Levrard. Optimal reach estimation and metric learning. The Annals of Statistics, 51(3):1086-1108, 2023-06. URL: https://doi.org/10.1214/23-AOS2281.
  2. Catherine Aaron and Olivier Bodart. Convergence rates for estimators of geodesic distances and Frèchet expectations. Journal of Applied Probability, 55(4):1001-1013, 2018-12. URL: https://doi.org/10.1017/jpr.2018.66.
  3. Ery Arias-Castro, Adel Javanmard, and Bruno Pelletier. Perturbation Bounds for Procrustes, Classical Scaling, and Trilateration, with Applications to Manifold Learning. Journal of Machine Learning Research, 21(1):498-534, 2020. URL: http://jmlr.org/papers/v21/18-720.html.
  4. Clément Berenfeld and Marc Hoffmann. Density estimation on an unknown submanifold. Electronic Journal of Statistics, 15(1):2179-2223, 2021-01. URL: https://doi.org/10.1214/21-EJS1826.
  5. Mira Bernstein, Vin Silva, John Langford, and Joshua Tenenbaum. Graph Approximations to Geodesics on Embedded Manifolds, 2000-12-20. URL: https://api.semanticscholar.org/CorpusID:10751942.
  6. Jean-Daniel Boissonnat, André Lieutier, and Mathijs Wintraecken. The reach, metric distortion, geodesic convexity and the variation of tangent spaces. Journal of Applied and Computational Topology, 3(1):29-58, 2019-06-01. URL: https://doi.org/10.1007/s41468-019-00029-8.
  7. Dmitri Burago, Yuri Burago, and Sergei Ivanov. A Course in Metric Geometry, volume 33 of Graduate Studies in Mathematics. American Mathematical Society, 2001-06-12. URL: https://doi.org/10.1090/gsm/033.
  8. Frédéric Chazal, David Cohen-Steiner, and Quentin Mérigot. Geometric Inference for Measures based on Distance Functions. Foundations of Computational Mathematics, 11(6):733, 2011. URL: https://doi.org/10.1007/s10208-011-9098-0.
  9. Frédéric Chazal, Pascal Massart, and Bertrand Michel. Rates of convergence for robust geometric inference. Electronic Journal of Statistics, 10(2):2243-2286, 2016-01. URL: https://doi.org/10.1214/16-EJS1161.
  10. Herbert Federer. Curvature measures. Transactions of the American Mathematical Society, 93(3):418-491, 1959. URL: https://doi.org/10.1090/S0002-9947-1959-0110078-1.
  11. Ximena Fernández, Eugenio Borghini, Gabriel Mindlin, and Pablo Groisman. Intrinsic persistent homology via density-based metric learning. Journal of Machine Learning Research, 24(1):75:3341-75:3382, 2023-01-01. URL: https://jmlr.org/papers/v24/21-1044.html.
  12. Nicolas Garcia Trillos, Anna Little, Daniel McKenzie, and James M. Murphy. Fermat Distances: Metric Approximation, Spectral Convergence, and Clustering Algorithms. Journal of Machine Learning Research, 25(1):176:8331-176:8395, 2024-01-01. URL: http://jmlr.org/papers/v25/23-0939.html.
  13. Pablo Groisman, Matthieu Jonckheere, and Facundo Sapienza. Nonhomogeneous Euclidean first-passage percolation and distance learning. Bernoulli, 28(1):255-276, 2022-02. URL: https://doi.org/10.3150/21-BEJ1341.
  14. Sung Jin Hwang, Steven B. Damelin, and Alfred O. Hero Iii. Shortest path through random points. The Annals of Applied Probability, 26(5):2791-2823, 2016-10. URL: https://doi.org/10.1214/15-AAP1162.
  15. André Lieutier and Mathijs Wintraecken. Manifolds of positive reach, differentiability, tangent variation, and attaining the reach, 2024-12-06. URL: https://doi.org/10.48550/arXiv.2412.04906.
  16. E. J. McShane. Extension of range of functions. Bulletin of the American Mathematical Society, 40(12):837-842, 1934. URL: https://doi.org/10.1090/S0002-9904-1934-05978-0.
  17. Jérôme Taupin. Estimation of Conformal Metrics. Google Scholar
  18. Jérôme Taupin and Frédéric Chazal. Fermat Distance-to-Measure: A robust Fermat-like metric, 2025-04-03. URL: https://arxiv.org/abs/2504.02381.
  19. Bin Yu. Assouad, Fano, and Le Cam. Research Papers in Probability and Statistics, 1997. URL: https://doi.org/10.1007/978-1-4612-1880-7_29.
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2026 Schloss Dagstuhl – LZI GmbH Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact