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Behaviour and Convergence of the Constrained Covariance


Technical Report


We discuss reproducing kernel Hilbert space (RKHS)-based measures of statistical dependence, with emphasis on constrained covariance (COCO), a novel criterion to test dependence of random variables. We show that COCO is a test for independence if and only if the associated RKHSs are universal. That said, no independence test exists that can distinguish dependent and independent random variables in all circumstances. Dependent random variables can result in a COCO which is arbitrarily close to zero when the source densities are highly non-smooth, which can make dependence hard to detect empirically. All current kernel-based independence tests share this behaviour. Finally, we demonstrate exponential convergence between the population and empirical COCO, which implies that COCO does not suffer from slow learning rates when used as a dependence test.

Author(s): Gretton, A. and Smola, A. and Bousquet, O. and Herbrich, R. and Schölkopf, B. and Logothetis, NK.
Number (issue): 130
Year: 2004
Day: 0

Department(s): Empirical Inference
Bibtex Type: Technical Report (techreport)

Institution: MPI for Biological Cybernetics

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

Links: PDF


  title = {Behaviour and Convergence of the Constrained Covariance},
  author = {Gretton, A. and Smola, A. and Bousquet, O. and Herbrich, R. and Sch{\"o}lkopf, B. and Logothetis, NK.},
  number = {130},
  organization = {Max-Planck-Gesellschaft},
  institution = {MPI for Biological Cybernetics},
  school = {Biologische Kybernetik},
  year = {2004}