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Kernel Methods for Measuring Independence




We introduce two new functionals, the constrained covariance and the kernel mutual information, to measure the degree of independence of random variables. These quantities are both based on the covariance between functions of the random variables in reproducing kernel Hilbert spaces (RKHSs). We prove that when the RKHSs are universal, both functionals are zero if and only if the random variables are pairwise independent. We also show that the kernel mutual information is an upper bound near independence on the Parzen window estimate of the mutual information. Analogous results apply for two correlation-based dependence functionals introduced earlier: we show the kernel canonical correlation and the kernel generalised variance to be independence measures for universal kernels, and prove the latter to be an upper bound on the mutual information near independence. The performance of the kernel dependence functionals in measuring independence is verified in the context of independent component analysis.

Author(s): Gretton, A. and Herbrich, R. and Smola, A. and Bousquet, O. and Schölkopf, B.
Journal: Journal of Machine Learning Research
Volume: 6
Pages: 2075-2129
Year: 2005
Month: December
Day: 0

Department(s): Empirical Inference
Bibtex Type: Article (article)

Digital: 0
Language: en
Organization: Max-Planck-Gesellschaft
School: Biologische Kybernetik

Links: PDF


  title = {Kernel Methods for Measuring Independence},
  author = {Gretton, A. and Herbrich, R. and Smola, A. and Bousquet, O. and Sch{\"o}lkopf, B.},
  journal = {Journal of Machine Learning Research},
  volume = {6},
  pages = {2075-2129},
  organization = {Max-Planck-Gesellschaft},
  school = {Biologische Kybernetik},
  month = dec,
  year = {2005},
  month_numeric = {12}