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Probability Theory and Related Fields

, Volume 171, Issue 3–4, pp 1045–1091 | Cite as

Convergence of the reach for a sequence of Gaussian-embedded manifolds

  • Robert J. Adler
  • Sunder Ram Krishnan
  • Jonathan E. Taylor
  • Shmuel Weinberger
Article
  • 58 Downloads

Abstract

Motivated by questions of manifold learning, we study a sequence of random manifolds, generated by embedding a fixed, compact manifold M into Euclidean spheres of increasing dimension via a sequence of Gaussian mappings. One of the fundamental smoothness parameters of manifold learning theorems is the reach, or critical radius, of M. Roughly speaking, the reach is a measure of a manifold’s departure from convexity, which incorporates both local curvature and global topology. This paper develops limit theory for the reach of a family of random, Gaussian-embedded, manifolds, establishing both almost sure convergence for the global reach, and a fluctuation theory for both it and its local version. The global reach converges to a constant well known both in the reproducing kernel Hilbert space theory of Gaussian processes, as well as in their extremal theory.

Keywords

Gaussian process Manifold Random embedding Critical radius Reach Curvature Asymptotics Fluctuation theory 

Mathematics Subject Classification

Primary 60G15 57N35 Secondary 60D05 60G60 

Notes

Acknowledgements

We would like to thank Takashi Owada for useful discussions, and two referees for helpful comments.

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Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Electrical EngineeringTechnionHaifaIsrael
  2. 2.Department of StatisticsStanford UniversityStanfordUSA
  3. 3.Department of MathematicsUniversity of ChicagoChicagoUSA

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