Geometric Jensen-Shannon Divergence Between Gaussian Measures On Hilbert Space

TL;DR

This paper introduces a closed-form expression for the Geometric Jensen-Shannon divergence between Gaussian measures on infinite-dimensional Hilbert spaces, generalizing finite-dimensional results and defining a regularized version applicable to all Gaussian pairs.

Contribution

It provides the first explicit formula for the divergence in infinite dimensions and introduces a regularized divergence that extends applicability beyond equivalent Gaussian measures.

Findings

Closed-form expression for divergence between equivalent Gaussian measures

Definition of a regularized divergence for all Gaussian measure pairs

Recovery of exact divergence in the limit of zero regularization

Abstract

This work studies the Geometric Jensen-Shannon divergence, based on the notion of geometric mean of probability measures, in the setting of Gaussian measures on an infinite-dimensional Hilbert space. On the set of all Gaussian measures equivalent to a fixed one, we present a closed form expression for this divergence that directly generalizes the finite-dimensional version. Using the notion of Log-Determinant divergences between positive definite unitized trace class operators, we then define a Regularized Geometric Jensen-Shannon divergence that is valid for any pair of Gaussian measures and that recovers the exact Geometric Jensen-Shannon divergence between two equivalent Gaussian measures when the regularization parameter tends to zero.