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Measuring statistical dependence with hilbert-schmidt norms
Arthur Gretton, Olivier Bousquet, Alex Smola and Bernhard Schölkopf
In: ALT 2005, October 2005, Singapore.
AbstractWe propose an independence criterion based on the eigen-
spectrum of covariance operators in reproducing kernel Hilbert spaces
(RKHSs), consisting of an empirical estimate of the Hilbert-Schmidt
norm of the cross-covariance operator (we term this a Hilbert-Schmidt In-
dependence Criterion, or HSIC). This approach has several advantages,
compared with previous kernel-based independence criteria. First, the
empirical estimate is simpler than any other kernel dependence test, and
requires no user-defined regularisation. Second, there is a clearly defined
population quantity which the empirical estimate approaches in the large
sample limit, with exponential convergence guaranteed between the two:
this ensures that independence tests based on HSIC do not suffer from
slow learning rates. Finally, we show in the context of independent com-
ponent analysis (ICA) that the performance of HSIC is competitive with
that of previously published kernel-based criteria, and of other recently
published ICA methods.
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