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Estimation of Conformal Metrics

Jérôme Taupin

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.

Estimation of Conformal 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(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.
Paper Structure (29 sections, 22 theorems, 122 equations, 2 figures)

This paper contains 29 sections, 22 theorems, 122 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Regularity of the Geodesics for the Conformal Metric
  3. Polygonal Approximation of the Conformal Metric
  4. Weighted Graphs
  5. Metric Approximation
  6. Unknown Conformal Factor
  7. Estimation from Random Samples
  8. Convergence of the Ball Graph Estimator
  9. Convergence of the Nearest-Neighbors Graph Estimator
  10. Minimax lower bound
  11. Case of a Smooth Manifold
  12. Practical Examples of Conformal Factors
  13. Density as a conformal factor
  14. Distance-to-measure as a conformal factor
  15. Conclusion
  16. ...and 14 more sections

Key Result

Lemma 2

boissonnatReachMetricDistortion2019 The reach of $M$ may be expressed as

Figures (2)

  • Figure 1: Bounding the angular velocity of a path with positive reach.
  • Figure 2: Carving an edge of the cube in $\mathbb{R}^3$.

Theorems & Definitions (28)

  • Definition 1
  • Lemma 2
  • Definition 3
  • Proposition 4
  • Proposition 5
  • Lemma 6
  • Definition 7
  • Definition 8
  • Proposition 9
  • Definition 10
  • ...and 18 more