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  Injective Hilbert Space Embeddings of Probability Measures

Sriperumbudur, B., Gretton, A., Fukumizu, K., Lanckriet, G., & Schölkopf, B. (2008). Injective Hilbert Space Embeddings of Probability Measures. In R. Servedio, & T. Zhang (Eds.), 21st Annual Conference on Learning Theory (COLT 2008) (pp. 111-122). Madison, WI, USA: Omnipress.

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Item Permalink: http://hdl.handle.net/11858/00-001M-0000-0013-C83D-C Version Permalink: http://hdl.handle.net/21.11116/0000-0003-3D85-3
Genre: Conference Paper

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 Creators:
Sriperumbudur, BK, Author              
Gretton, A1, 2, Author              
Fukumizu, K, Author              
Lanckriet, G, Author
Schölkopf, B1, 2, Author              
Affiliations:
1Department Empirical Inference, Max Planck Institute for Biological Cybernetics, Max Planck Society, ou_1497795              
2Max Planck Institute for Biological Cybernetics, Max Planck Society, Spemannstrasse 38, 72076 Tübingen, DE, ou_1497794              

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 Abstract: 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.

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 Dates: 2008-07
 Publication Status: Published in print
 Pages: -
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 Table of Contents: -
 Rev. Type: -
 Identifiers: BibTex Citekey: 5122
 Degree: -

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Title: 21st Annual Conference on Learning Theory (COLT 2008)
Place of Event: Helsinki, Finland
Start-/End Date: 2008-07-09 - 2008-07-12

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Title: 21st Annual Conference on Learning Theory (COLT 2008)
Source Genre: Proceedings
 Creator(s):
Servedio, RA, Editor
Zhang, T, Editor
Affiliations:
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Publ. Info: Madison, WI, USA : Omnipress
Pages: - Volume / Issue: - Sequence Number: - Start / End Page: 111 - 122 Identifier: ISBN: 978-1-60558-205-4