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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
  • 117 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.

References

  1. 1.
    Adler, R.J., Taylor, J.E.: Random Fields and Geometry. Springer Monographs in Mathematics. Springer, New York (2007)Google Scholar
  2. 2.
    Amelunxen, D., Bürgisser, P.: Probabilistic analysis of the Grassmann condition number. arXiv:1112.2603
  3. 3.
    Amenta, N., Bern, M.: Surface reconstruction by Voronoi filtering. Discrete Comput. Geom. 22(4), 481–504 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Anderson, T.W.: An Introduction to Multivariate Statistical Analysis. Wiley Series in Probability and Statistics, 3rd edn. Wiley, Hoboken (2003)Google Scholar
  5. 5.
    Billingsley, P.: Convergence of Probability Measures. Wiley Series in Probability and Statistics: Probability and Statistics, 2nd edn. Wiley, New York (1999)CrossRefzbMATHGoogle Scholar
  6. 6.
    Boissonnat, J.-D., Chazal, F., Yvinec, M.: Computational geometry and topology for data analysis. Book in preparation. http://geometrica.saclay.inria.fr/team/Fred.Chazal/papers/CGLcourseNotes/main.pdf (2016)
  7. 7.
    Cartwright, D.I., Field, M.J.: A refinement of the arithmetic mean-geometric mean inequality. Proc. Am. Math. Soc. 71(1), 36–38 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chazal, F., Lieutier, A.: Smooth manifold reconstruction from noisy and non-uniform approximation with guarantees. Comput. Geom. 40(2), 156–170 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Federer, H.: Curvature measures. Trans. Am. Math. Soc. 93, 418–491 (1959)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Gray, A.: Tubes. Advanced Book Program. Addison-Wesley Publishing Company, Redwood City (1990)Google Scholar
  11. 11.
    Hotelling, H.: Tubes and spheres in \(n\)-spaces and a class of statistical problems. Am. J. Math. 61, 440–460 (1939)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Johansen, S., Johnstone, I.M.: Hotelling’s theorem on the volume of tubes: some illustrations in simultaneous inference and data analysis. Ann. Statist. 18(2), 652–684 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kendall, M., Stuart, A.: The Advanced Theory of Statistics, Design and Analysis, and Time-Series, vol. 3, 3rd edn. Hafner Press [Macmillan Publishing Co., Inc.], New York (1976)zbMATHGoogle Scholar
  14. 14.
    Kendall, M.G., Stuart, A.: The Advanced Theory of Statistics. Distribution Theory, vol. 1, 3rd edn. Hafner Publishing Co., New York (1969)zbMATHGoogle Scholar
  15. 15.
    Krishnan, S.R., Taylor, J.E., Adler, R.J.: The intrinsic geometry of some random manifolds. arXiv:1512.05622 (2015)
  16. 16.
    Ledoux, M., Talagrand, M.: Probability in Banach spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Isoperimetry and Processes. Springer, Berlin (1991)Google Scholar
  17. 17.
    Lee, J.M.: Riemannian Manifolds, An Introduction to Curvature, Volume 176 of Graduate Texts in Mathematics, vol. 176. Springer, New York (1997)Google Scholar
  18. 18.
    Niyogi, P., Smale, S., Weinberger, S.: A topological view of unsupervised learning from noisy data. SIAM J. Comput. 40(3), 646–663 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Niyogi, P., Smale, S., Weinberger, S.: Finding the homology of submanifolds with high confidence from random samples. Discrete Comput. Geom. 39(1–3), 419–441 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Olkin, I., Pratt, J.W.: Unbiased estimation of certain correlation coefficients. Ann. Math. Statist 29, 201–211 (1958)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Smale, S.: Complexity theory and numerical analysis. In: Iserles, A. (ed.) Acta Numerica, vol. 6, pp. 523–551. Cambridge University Press, Cambridge (1997)Google Scholar
  22. 22.
    Takemura, A., Kuriki, S.: On the equivalence of the tube and Euler characteristic methods for the distribution of the maximum of Gaussian fields over piecewise smooth domains. Ann. Appl. Probab. 12(2), 768–796 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Takemura, A., Kuriki, S.: Tail probability via the tube formula when the critical radius is zero. Bernoulli 9(3), 535–558 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Taylor, J., Takemura, A., Adler, R.J.: Validity of the expected Euler characteristic heuristic. Ann. Probab. 33(4), 1362–1396 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Thäle, C.: 50 years sets with positive reach—a survey. Surv. Math. Appl. 3, 123–165 (2008)MathSciNetzbMATHGoogle Scholar
  26. 26.
    van der Vaart, A.W., Wellner, J.A.: Weak Convergence and Empirical Processes. Springer Series in Statistics. With Applications to Statistics. Springer, New York (1996)zbMATHGoogle Scholar
  27. 27.
    Weyl, H.: On the volume of tubes. Am. J. Math. 61(2), 461–472 (1939)MathSciNetCrossRefzbMATHGoogle Scholar

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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