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Injective Hilbert Space Embeddings of Probability Measures

2008

Conference Paper

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A Hilbert space embedding for probability measures has recently been proposed, with applications including dimensionality reduction, homogeneity testing and independence testing. This embedding represents any probability measure as a mean element in a reproducing kernel Hilbert space (RKHS). The embedding function has been proven to be injective when the reproducing kernel is universal. In this case, the embedding induces a metric on the space of probability distributions defined on compact metric spaces. In the present work, we consider more broadly the problem of specifying characteristic kernels, defined as kernels for which the RKHS embedding of probability measures is injective. In particular, characteristic kernels can include non-universal kernels. We restrict ourselves to translation-invariant kernels on Euclidean space, and define the associated metric on probability measures in terms of the Fourier spectrum of the kernel and characteristic functions of these measures. The support of the kernel spectrum is important in finding whether a kernel is characteristic: in particular, the embedding is injective if and only if the kernel spectrum has the entire domain as its support. Characteristic kernels may nonetheless have difficulty in distinguishing certain distributions on the basis of finite samples, again due to the interaction of the kernel spectrum and the characteristic functions of the measures.

Author(s): Sriperumbudur, BK. and Gretton, A. and Fukumizu, K. and Lanckriet, G. and Schölkopf, B.
Book Title: Proceedings of the 21st Annual Conference on Learning Theory
Journal: Proceedings of the 21st Annual Conference on Learning Theory (COLT 2008)
Pages: 111-122
Year: 2008
Month: July
Day: 0
Editors: RA Servedio and T Zhang
Publisher: Omnipress

Department(s): Empirical Inference
Bibtex Type: Conference Paper (inproceedings)

Address: Madison, WI, USA
Digital: 0
Event Name: 21st Annual Conference on Learning Theory (COLT 2008)
Event Place: Helsinki, Finland
Language: en
Organization: Max-Planck-Gesellschaft
School: Biologische Kybernetik

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BibTex

@inproceedings{5122,
  title = {Injective Hilbert Space Embeddings of Probability Measures},
  author = {Sriperumbudur, BK. and Gretton, A. and Fukumizu, K. and Lanckriet, G. and Sch{\"o}lkopf, B.},
  journal = {Proceedings of the 21st Annual Conference on Learning Theory (COLT 2008)},
  booktitle = {Proceedings of the 21st Annual Conference on Learning Theory},
  pages = {111-122},
  editors = {RA Servedio and T Zhang},
  publisher = {Omnipress},
  organization = {Max-Planck-Gesellschaft},
  school = {Biologische Kybernetik},
  address = {Madison, WI, USA},
  month = jul,
  year = {2008},
  month_numeric = {7}
}