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Kernel Choice and Classifiability for RKHS Embeddings of Probability Distributions

2009

Conference Paper

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Embeddings of probability measures into reproducing kernel Hilbert spaces have been proposed as a straightforward and practical means of representing and comparing probabilities. In particular, the distance between embeddings (the maximum mean discrepancy, or MMD) has several key advantages over many classical metrics on distributions, namely easy computability, fast convergence and low bias of finite sample estimates. An important requirement of the embedding RKHS is that it be characteristic: in this case, the MMD between two distributions is zero if and only if the distributions coincide. Three new results on the MMD are introduced in the present study. First, it is established that MMD corresponds to the optimal risk of a kernel classifier, thus forming a natural link between the distance between distributions and their ease of classification. An important consequence is that a kernel must be characteristic to guarantee classifiability between distributions in the RKHS. Second, the class of characteristic kernels is broadened to incorporate all strictly positive definite kernels: these include non-translation invariant kernels and kernels on non-compact domains. Third, a generalization of the MMD is proposed for families of kernels, as the supremum over MMDs on a class of kernels (for instance the Gaussian kernels with different bandwidths). This extension is necessary to obtain a single distance measure if a large selection or class of characteristic kernels is potentially appropriate. This generalization is reasonable, given that it corresponds to the problem of learning the kernel by minimizing the risk of the corresponding kernel classifier. The generalized MMD is shown to have consistent finite sample estimates, and its performance is demonstrated on a homogeneity testing example.

Author(s): Sriperumbudur, BK. and Fukumizu, K. and Gretton, A. and Lanckriet, GRG. and Schölkopf, B.
Book Title: Advances in Neural Information Processing Systems 22
Journal: Advances in Neural Information Processing Systems 22: 23rd Annual Conference on Neural Information Processing Systems 2009
Pages: 1750-1758
Year: 2009
Day: 0
Editors: Y Bengio and D Schuurmans and J Lafferty and C Williams and A Culotta
Publisher: Curran

Department(s): Empirical Inference
Research Project(s): Kernel Methods
Bibtex Type: Conference Paper (inproceedings)

Event Name: 23rd Annual Conference on Neural Information Processing Systems (NIPS 2009)
Event Place: Vancouver, BC, Canada

Address: Red Hook, NY, USA
Digital: 0
ISBN: 978-1-615-67911-9
Language: en
Organization: Max-Planck-Gesellschaft
School: Biologische Kybernetik

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BibTex

@inproceedings{6131,
  title = {Kernel Choice and Classifiability for RKHS Embeddings of Probability Distributions},
  author = {Sriperumbudur, BK. and Fukumizu, K. and Gretton, A. and Lanckriet, GRG. and Sch{\"o}lkopf, B.},
  journal = {Advances in Neural Information Processing Systems 22: 23rd Annual Conference on Neural Information Processing Systems 2009},
  booktitle = {Advances in Neural Information Processing Systems 22},
  pages = {1750-1758},
  editors = {Y Bengio and D Schuurmans and J Lafferty and C Williams and A Culotta},
  publisher = {Curran},
  organization = {Max-Planck-Gesellschaft},
  school = {Biologische Kybernetik},
  address = {Red Hook, NY, USA},
  year = {2009}
}