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Behaviour and Convergence of the Constrained Covariance

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Gretton,  A
Department Empirical Inference, Max Planck Institute for Biological Cybernetics, Max Planck Society;
Max Planck Institute for Biological Cybernetics, Max Planck Society;

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Bousquet,  O
Department Empirical Inference, Max Planck Institute for Biological Cybernetics, Max Planck Society;
Max Planck Institute for Biological Cybernetics, Max Planck Society;

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Schölkopf,  B
Department Empirical Inference, Max Planck Institute for Biological Cybernetics, Max Planck Society;
Max Planck Institute for Biological Cybernetics, Max Planck Society;

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Logothetis,  NK
Department Physiology of Cognitive Processes, Max Planck Institute for Biological Cybernetics, Max Planck Society;
Max Planck Institute for Biological Cybernetics, Max Planck Society;

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MPIK-TR-130.pdf
(Publisher version), 348KB

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Citation

Gretton, A., Smola, A., Bousquet, O., Herbrich, R., Schölkopf, B., & Logothetis, N.(2004). Behaviour and Convergence of the Constrained Covariance (130). Tübingen, Germany: Max Planck Institute for Biological Cybernetics.


Cite as: https://hdl.handle.net/11858/00-001M-0000-0013-F353-1
Abstract
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.