Information Geometry for Wasserstein KL Divergence of Gaussian Measures on $\mathbb{R}^n$

Abstract

We study the Wasserstein Kullback--Leibler divergence (WKL divergence) on the manifold of nondegenerate Gaussian measures over $\mathbb R^n$. In the canonical-divergence construction, the Fisher--Rao metric recovers forward KL along intrinsic mixture geodesics and reverse KL along geodesics of the conjugate exponential connection. Replacing the Fisher--Rao metric by the Otto metric and following the latter route produces the $e_1$-connection underlying WKL divergence. We establish its geodesic completeness, classify its forward limits, prove that every ordered pair is joined by a unique $e_1$-connector generated by a quadratic potential, and derive an explicit WKL divergence formula with separate mean and covariance contributions. WKL divergence is nonnegative and separating. At equal covariances WKL divergence equals one half of the squared Euclidean mean distance. Finally, WKL divergence extends finitely and continuously to singular targets from a nondegenerate source, but diverges when the target remains nondegenerate and the source covariance becomes singular. Path-dependent joint limits at Dirac pairs preclude a continuous extension to the full positive-semidefinite covariance product, although a lower-semicontinuous extended-real extension exists.