The Representation Jensen-R\'enyi Divergence

TL;DR

This paper introduces a new divergence measure between data distributions based on kernel operators, which avoids explicit distribution estimation and demonstrates strong empirical performance in distribution comparison and classification tasks.

Contribution

It proposes a novel Jensen-Rényi divergence using kernel eigenvalues, with proven convergence properties and practical advantages over traditional distribution estimation methods.

Findings

Achieves state-of-the-art results in distribution comparison

Provides a divergence measure with favorable theoretical properties

Effective in sampling unbalanced data for classification

Abstract

We introduce a divergence measure between data distributions based on operators in reproducing kernel Hilbert spaces defined by kernels. The empirical estimator of the divergence is computed using the eigenvalues of positive definite Gram matrices that are obtained by evaluating the kernel over pairs of data points. The new measure shares similar properties to Jensen-Shannon divergence. Convergence of the proposed estimators follows from concentration results based on the difference between the ordered spectrum of the Gram matrices and the integral operators associated with the population quantities. The proposed measure of divergence avoids the estimation of the probability distribution underlying the data. Numerical experiments involving comparing distributions and applications to sampling unbalanced data for classification show that the proposed divergence can achieve state of the art results.