A Reproducing Kernel Hilbert Space Framework for Information-Theoretic Learning

This paper provides a functional analysis perspective of information-theoretic learning (ITL) by defining bottom-up a reproducing kernel Hilbert space (RKHS) uniquely determined by the symmetric nonnegative definite kernel function known as the cross-information potential (CIP). The CIP as an integral of the product of two probability density functions characterizes similarity between two stochastic functions. We prove the existence of a one-to-one congruence mapping between the ITL RKHS and the Hilbert space spanned by square integrable probability density functions. Therefore, all the statistical descriptors in the original information-theoretic learning formulation can be rewritten as algebraic computations on deterministic functional vectors in the ITL RKHS, instead of limiting the functional view to the estimators as is commonly done in kernel methods. A connection between the ITL RKHS and kernel approaches interested in quantifying the statistics of the projected data is also established.