Estimation of Conformal Metrics
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910025781346304 |
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| author | Taupin, Jérôme |
| author_facet | Taupin, Jérôme |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_16466 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Estimation of Conformal Metrics Taupin, Jérôme Statistics Theory 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. |
| title | Estimation of Conformal Metrics |
| topic | Statistics Theory |
| url | https://arxiv.org/abs/2602.16466 |