The Representation Jensen-Rényi Divergence
This work addresses the problem of comparing data distributions without probability estimation, which is useful for machine learning practitioners dealing with unbalanced datasets, though it appears incremental as it builds on existing divergence concepts.
The authors introduced a new divergence measure between data distributions using kernel-based operators, which avoids estimating underlying probability distributions and achieved state-of-the-art results in numerical experiments for distribution comparison and sampling unbalanced data.
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.