Kernel methods for testing independence
Arthur Gretton, Ralf Herbrich, Alex Smola, Olivier Bousquet and Bernhard Schölkopf
Journal of Machine Learning Research
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 indepen-
dence 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.