Density Estimators of Gaussian Type on Closed Riemannian Manifolds

Abstract

We prove consistency results for two types of density estimators on a closed, connected Riemannian manifold under suitable regularity conditions. The convergence rates are consistent with those in Euclidean space as well as those obtained for a previously proposed class of kernel density estimators on closed Riemannian manifolds. The first estimator is the uniform mixture of heat kernels centered at each observation, a natural extension of the usual Gaussian estimator to Riemannian manifolds. The second is an approximate heat kernel (AHK) estimator that is motivated by more practical considerations, where observations occur on a manifold isometrically embedded in Euclidean space whose structure or heat kernel may not be completely known. We also provide some numerical evidence that the predicted convergence rate is attained for the AHK estimator.

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Notes

  1. 1.

    This assumption is not restrictive, as the Nash embedding theorem shows.

  2. 2.

    We used Matlab code written by Rob Parrish, The Sherrill Group, CCMST Georgia Tech.

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Acknowledgements

We thank Misha Belkin for discussions and comments about the paper.

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Correspondence to Washington Mio.

Additional information

This research was supported in part by NSF grant DBI-1052942.

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Bates, J., Mio, W. Density Estimators of Gaussian Type on Closed Riemannian Manifolds. J Math Imaging Vis 50, 53–59 (2014). https://doi.org/10.1007/s10851-013-0460-5

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Keywords

  • Density estimation
  • Heat kernel
  • Distributions on Riemannian manifolds