Statistical and Geometrical properties of regularized Kernel Kullback-Leibler divergence

TL;DR

This paper introduces a regularized version of the kernel Kullback-Leibler divergence (KKL) that is well-defined for all distributions, provides theoretical bounds, and develops practical algorithms for discrete distributions.

Contribution

It proposes a regularized KKL divergence, derives bounds and a closed-form expression, and develops a Wasserstein gradient descent scheme for discrete distributions.

Findings

Regularized KKL is well-defined for all distributions.

Finite-sample bounds for the divergence are established.

A practical gradient descent algorithm for discrete distributions is demonstrated.

Abstract

In this paper, we study the statistical and geometrical properties of the Kullback-Leibler divergence with kernel covariance operators (KKL) introduced by Bach [2022]. Unlike the classical Kullback-Leibler (KL) divergence that involves density ratios, the KKL compares probability distributions through covariance operators (embeddings) in a reproducible kernel Hilbert space (RKHS), and compute the Kullback-Leibler quantum divergence. This novel divergence hence shares parallel but different aspects with both the standard Kullback-Leibler between probability distributions and kernel embeddings metrics such as the maximum mean discrepancy. A limitation faced with the original KKL divergence is its inability to be defined for distributions with disjoint supports. To solve this problem, we propose in this paper a regularised variant that guarantees that the divergence is well defined for all distributions. We derive bounds that quantify the deviation of the regularised KKL to the original one, as well as finite-sample bounds. In addition, we provide a closed-form expression for the regularised KKL, specifically applicable when the distributions consist of finite sets of points, which makes it implementable. Furthermore, we derive a Wasserstein gradient descent scheme of the KKL divergence in the case of discrete distributions, and study empirically its properties to transport a set of points to a target distribution.