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Kernel Density Estimation on Symmetric Spaces

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Geometric Science of Information (GSI 2015)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 9389))

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Abstract

We introduce a novel kernel density estimator for a large class of symmetric spaces and prove a minimax rate of convergence as fast as the minimax rate on Euclidean space. We prove a minimax rate of convergence proven without any compactness assumptions on the space or Hölder-class assumptions on the densities. A main tool used in proving the convergence rate is the Helgason-Fourier transform, a generalization of the Fourier transform for semisimple Lie groups modulo maximal compact subgroups. This paper obtains a simplified formula in the special case when the symmetric space is the 2-dimensional hyperboloid.

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References

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Acknowledgements

This work was partially supported by an NSF Graduate Research Fellowship under grant DGE-1252522. Also, the author is grateful to her advisor, Cosma Shalizi, for his invaluable guidance and feedback throughout this research.

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Authors and Affiliations

  1. Carnegie Mellon University, Pittsburgh, PA, 15217, USA

    Dena Marie Asta

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  1. Dena Marie Asta
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Correspondence to Dena Marie Asta .

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Editors and Affiliations

  1. Bâtiment Alan Turing, CS35003, École Polytechnique, Palaiseau, France

    Frank Nielsen

  2. Thales Land\& Air Systems, Limours, France

    Frédéric Barbaresco

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Asta, D.M. (2015). Kernel Density Estimation on Symmetric Spaces. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2015. Lecture Notes in Computer Science(), vol 9389. Springer, Cham. https://doi.org/10.1007/978-3-319-25040-3_83

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  • DOI: https://doi.org/10.1007/978-3-319-25040-3_83

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-25039-7

  • Online ISBN: 978-3-319-25040-3

  • eBook Packages: Computer ScienceComputer Science (R0)

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