Statistical divergences quantify the difference between probability distributions finding multiple uses in machine-learning. However, a fundamental challenge is to estimate divergence from empirical samples since the underlying distributions of the data are usually unknown. In this work, we propose the representation Jensen-Shannon Divergence, a novel divergence based on covariance operators in reproducing kernel Hilbert spaces (RKHS). Our approach embeds the data distributions in an RKHS and exploits the spectrum of the covariance operators of the representations. We provide an estimator from empirical covariance matrices by explicitly mapping the data to an RKHS using Fourier features. This estimator is flexible, scalable, differentiable, and suitable for minibatch-based optimization problems. Additionally, we provide an estimator based on kernel matrices without having an explicit mapping to the RKHS. We show that this quantity is a lower bound on the Jensen-Shannon divergence, and we propose a variational approach to estimate it. We applied our divergence to two-sample testing outperforming related state-of-the-art techniques in several datasets. We used the representation Jensen-Shannon divergence as a cost function to train generative adversarial networks which intrinsically avoids mode collapse and encourages diversity.
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