Entropy-Regularized $2$-Wasserstein Distance between Gaussian Measures
This provides theoretical tools for Gaussian-based uncertainty quantification, but is incremental as it extends existing Wasserstein frameworks to a regularized setting.
The paper derived closed-form expressions for the entropy-regularized 2-Wasserstein distance between Gaussian measures, including interpolations and barycenters, and analyzed limiting cases of the Sinkhorn divergence.
Gaussian distributions are plentiful in applications dealing in uncertainty quantification and diffusivity. They furthermore stand as important special cases for frameworks providing geometries for probability measures, as the resulting geometry on Gaussians is often expressible in closed-form under the frameworks. In this work, we study the Gaussian geometry under the entropy-regularized 2-Wasserstein distance, by providing closed-form solutions for the distance and interpolations between elements. Furthermore, we provide a fixed-point characterization of a population barycenter when restricted to the manifold of Gaussians, which allows computations through the fixed-point iteration algorithm. As a consequence, the results yield closed-form expressions for the 2-Sinkhorn divergence. As the geometries change by varying the regularization magnitude, we study the limiting cases of vanishing and infinite magnitudes, reconfirming well-known results on the limits of the Sinkhorn divergence. Finally, we illustrate the resulting geometries with a numerical study.