Kernel Constrained Covariance for Dependence Measurement
Arthur Gretton, Alex Smola, Olivier Bousquet, Ralf Herbrich, Andrei Belitski, Mark Augath, Yusuke Murayama, John Pauls, Bernhard Schölkopf and Nikos Logothetis
In: AISTATS 2005, January 6-8, 2005, Barbados.
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
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. All current kernel-based independence tests share this behaviour. We demonstrate exponential convergence between the population and empirical COCO. Finally, we use COCO as a measure of joint neural activity between voxels in MRI recordings of the macaque monkey, and compare the results to the mutual information and the correlation. We also show the effect of removing breathing artefacts from the MRI recording.