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Approximate Quadrature Measures on Data-Defined Spaces

  • Hrushikesh N. MhaskarEmail author
Chapter

Abstract

An important question in the theory of approximate integration is to study the conditions on the nodes xk,n and weights wk,n that allow an estimate of the form
$$\displaystyle \sup _{f\in \mathcal {B}_\gamma }\left |\sum _k w_{k,n}{\,f}(x_{k,n})-\int _{\mathbb {X}} fd\mu ^*\right | \le cn^{-\gamma }, \qquad n=1,2,\cdots , $$
where \(\mathbb {X}\) is often a manifold with its volume measure μ, and \(\mathcal {B}_\gamma \) is the unit ball of a suitably defined smoothness class, parametrized by γ. In this paper, we study this question in the context of a quasi-metric, locally compact, measure space \(\mathbb {X}\) with a probability measure μ. We show that quadrature formulas exact for integrating the so called diffusion polynomials of degree < n satisfy such estimates. Without requiring exactness, such formulas can be obtained as a solutions of some kernel-based optimization problem. We discuss the connection with the question of optimal covering radius. Our results generalize in some sense many recent results in this direction.

Notes

Acknowledgements

The research of this author is supported in part by ARO Grant W911NF-15-1-0385. We thank the referees for their careful reading and helpful comments.

References

  1. 1.
    Belkin, M., Niyogi, P.: Semi-supervised learning on Riemannian manifolds. Mach. Learn. 56(1–3), 209–239 (2004)CrossRefGoogle Scholar
  2. 2.
    Belkin, M., Niyogi, P.: Convergence of Laplacian eigenmaps. Adv. Neural Inf. Proces. Syst. 19, 129 (2007)Google Scholar
  3. 3.
    Belkin, M., Niyogi, P.: Towards a theoretical foundation for Laplacian-based manifold methods. J. Comput. Syst. Sci. 74(8), 1289–1308 (2008)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bondarenko, A., Radchenko, D., Viazovska, M.: Optimal asymptotic bounds for spherical designs. Ann. Math. 178(2), 443–452 (2013)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Brandolini, L., Choirat, C., Colzani, L., Gigante, G., Seri, R., Travaglini, G.: Quadrature rules and distribution of points on manifolds. Ann. Sc. Norm. Super. Pisa Cl. Sci 13, 889–923 (2014)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Brauchart, J., Saff, E., Sloan, I., Womersley, R.: QMC designs: optimal order quasi monte carlo integration schemes on the sphere. Math. Comput. 83(290), 2821–2851 (2014)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Brauchart, J.S., Dick, J., Saff, E.B., Sloan, I.H., Wang, Y.G., Womersley, R.S.: Covering of spheres by spherical caps and worst-case error for equal weight cubature in Sobolev spaces. J. Math. Anal. Appl. 431(2), 782–811 (2015)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Breger, A., Ehler, M., Graef, M.: Points on manifolds with asymptotically optimal covering radius. Preprint at arXiv:1607.06899 (2016)Google Scholar
  9. 9.
    Breger, A., Ehler, M., Graef, M.: Quasi monte carlo integration and kernel-based function approximation on Grassmannians. Preprint at arXiv:1605.09165 (2016)Google Scholar
  10. 10.
    Coifman, R.R., Lafon, S.: Diffusion maps. Appl. Comput. Harmon. Anal. 21(1), 5–30 (2006)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Davies, E.B.: L p spectral theory of higher-order elliptic differential operators. Bull. Lond. Math. Soc. 29(05), 513–546 (1997)MathSciNetCrossRefGoogle Scholar
  12. 12.
    DeVore, R.A., Lorentz, G.G.: Constructive Approximation, vol. 303. Springer Science & Business Media, Berlin (1993)zbMATHGoogle Scholar
  13. 13.
    Ehler, M., Filbir, F., Mhaskar, H.N.: Locally learning biomedical data using diffusion frames. J. Comput. Biol. 19(11), 1251–1264 (2012)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Filbir, F., Mhaskar, H.N.: A quadrature formula for diffusion polynomials corresponding to a generalized heat kernel. J. Fourier Anal. Appl. 16(5), 629–657 (2010)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Filbir, F., Mhaskar, H.N.: Marcinkiewicz–Zygmund measures on manifolds. J. Complex. 27(6), 568–596 (2011)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Geller, D., Pesenson, I.Z.: Band-limited localized Parseval frames and Besov spaces on compact homogeneous manifolds. J. Geom. Anal. 21(2), 334–371 (2011)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Grigor’yan, A.: Estimates of heat kernels on Riemannian manifolds. Lond. Math. Soc. Lect. Note Ser. 273, 140–225 (1999)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Grigor’yan, A.: Heat kernels on metric measure spaces with regular volume growth. In: Handbook of Geometric Analysis, no. 2. Adv. Lect. Math. (ALM), vol. 13, pp. 1–60. Int. Press, Somerville (2010)Google Scholar
  19. 19.
    Grigor’yan, A.: Heat kernels on weighted manifolds and applications. Contemp. Math 398, 93–191 (2006)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Hesse, K., Mhaskar, H.N., Sloan, I.H.: Quadrature in Besov spaces on the Euclidean sphere. J. Complex. 23(4), 528–552 (2007)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Jones, P.W., Maggioni, M., Schul, R.: Manifold parametrizations by eigenfunctions of the Laplacian and heat kernels. Proc. Natl. Acad. Sci. 105(6), 1803–1808 (2008)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Kordyukov, Y.A.: L p-theory of elliptic differential operators on manifolds of bounded geometry. Acta Appl. Math. 23(3), 223–260 (1991)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Lafon, S.S.: Diffusion maps and geometric harmonics. Ph.D. thesis, Yale University (2004)Google Scholar
  24. 24.
    Maggioni, M., Mhaskar, H.N.: Diffusion polynomial frames on metric measure spaces. Appl. Comput. Harmon. Anal. 24(3), 329–353 (2008)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Mhaskar, H.N.: Eignets for function approximation on manifolds. Appl. Comput. Harmon. Anal. 29(1), 63–87 (2010)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Mhaskar, H.N.: A generalized diffusion frame for parsimonious representation of functions on data defined manifolds. Neural Netw. 24(4), 345–359 (2011)CrossRefGoogle Scholar
  27. 27.
    Mhaskar, H.N.: A unified framework for harmonic analysis of functions on directed graphs and changing data. Appl. Comput. Harmon. Anal. Published online June 28 (2016)Google Scholar
  28. 28.
    Rivlin, T.J.: The Chebyshev Polynomials. Wiley, New York (1974)zbMATHGoogle Scholar
  29. 29.
    Rosasco, L., Belkin, M., Vito, E.D.: On learning with integral operators. J. Mach. Learn. Res. 11, 905–934 (2010)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Singer, A.: From graph to manifold Laplacian: the convergence rate. Appl. Comput. Harmon. Anal. 21(1), 128–134 (2006)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of Mathematical SciencesClaremont Graduate UniversityClaremontUSA

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