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A Kernel Two-Sample Test

2012

Article

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We propose a framework for analyzing and comparing distributions, which we use to construct statistical tests to determine if two samples are drawn from different distributions. Our test statistic is the largest difference in expectations over functions in the unit ball of a reproducing kernel Hilbert space (RKHS), and is called the maximum mean discrepancy (MMD). We present two distribution-free tests based on large deviation bounds for the MMD, and a third test based on the asymptotic distribution of this statistic. The MMD can be computed in quadratic time, although efficient linear time approximations are available. Our statistic is an instance of an integral probability metric, and various classical metrics on distributions are obtained when alternative function classes are used in place of an RKHS. We apply our two-sample tests to a variety of problems, including attribute matching for databases using the Hungarian marriage method, where they perform strongly. Excellent performance is also obtained when comparing distributions over graphs, for which these are the first such tests.

Author(s): Gretton, A. and Borgwardt, K. and Rasch, M. and Schölkopf, B. and Smola, A.
Journal: Journal of Machine Learning Research
Volume: 13
Pages: 723--773
Year: 2012
Month: March
Day: 0

Department(s): Empirical Inference
Research Project(s): Kernel Methods
Bibtex Type: Article (article)

Digital: 0

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BibTex

@article{Scholkopf2012,
  title = {A Kernel Two-Sample Test },
  author = {Gretton, A. and Borgwardt, K. and Rasch, M. and Sch{\"o}lkopf, B. and Smola, A.},
  journal = {Journal of Machine Learning Research},
  volume = {13},
  pages = {723--773},
  month = mar,
  year = {2012},
  month_numeric = {3}
}