# Entropy-Regularized 2-Wasserstein Distance between Gaussian Measures

arXiv: https://arxiv.org/abs/2006.03416 · HTML: https://mlsanity.com/paper/2006.03416/

## Key facts

- Cited by 32 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Jun 2020.

## Built on by (most cited first, 30 of 32)

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- [The Schr"odinger Bridge between Gaussian Measures has a Closed Form](https://mlsanity.com/paper/2202.05722/) (2022)
- [Averaging on the Bures-Wasserstein manifold: dimension-free convergence of gradient descent](https://mlsanity.com/paper/2106.08502/) (2021)
- [Scalable Optimal Transport Methods in Machine Learning: A Contemporary Survey](https://mlsanity.com/paper/2305.05080/) (2023)
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- [An improved central limit theorem and fast convergence rates for entropic transportation costs](https://mlsanity.com/paper/2204.09105/) (2022)
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- [Accelerating Sinkhorn Algorithm with Sparse Newton Iterations](https://mlsanity.com/paper/2401.12253/) (2024)
- [Doubly Regularized Entropic Wasserstein Barycenters](https://mlsanity.com/paper/2303.11844/) (2023)
- [Critical Points and Convergence Analysis of Generative Deep Linear Networks Trained with Bures-Wasserstein Loss](https://mlsanity.com/paper/2303.03027/) (2023)
- [Go With the Flow: Fast Diffusion for Gaussian Mixture Models](https://mlsanity.com/paper/2412.09059/) (2024)
- [BM2: Coupled Schr"odinger Bridge Matching](https://mlsanity.com/paper/2409.09376/) (2024)
- [Topological Schr"odinger Bridge Matching](https://mlsanity.com/paper/2504.04799/) (2025)
- [Convergence and finite sample approximations of entropic regularized Wasserstein distances in Gaussian and RKHS settings](https://mlsanity.com/paper/2101.01429/) (2021)
- [Wasserstein Distributionally Robust Optimization with Wasserstein Barycenters](https://mlsanity.com/paper/2203.12136/) (2022)
- [Learning non-equilibrium diffusions with Schr"odinger bridges: from exactly solvable to simulation-free](https://mlsanity.com/paper/2505.16644/) (2025)
- [Entropic optimal transport beyond product reference couplings: the Gaussian case on Euclidean space](https://mlsanity.com/paper/2507.01709/) (2025)
- [Robust inference using density-powered Stein operators](https://mlsanity.com/paper/2511.03963/) (2025)
- [Estimating 2-Sinkhorn Divergence between Gaussian Processes from Finite-Dimensional Marginals](https://mlsanity.com/paper/2102.03267/) (2021)
- [An Optimal Transport Approach for Network Regression](https://mlsanity.com/paper/2406.12204/) (2024)
- [Information Geometry for Wasserstein KL Divergence of Gaussian Measures on ℝn](https://mlsanity.com/paper/2503.24022/) (2025)
- [New Trends in the Stability of Sinkhorn Semigroups](https://mlsanity.com/paper/2601.12633/) (2026)
- [Finite sample approximations of exact and entropic Wasserstein distances between covariance operators and Gaussian processes](https://mlsanity.com/paper/2104.12368/) (2021)
- [Computational Guarantees for Doubly Entropic Wasserstein Barycenters via Damped Sinkhorn Iterations](https://mlsanity.com/paper/2307.13370/) (2023)
- [Optimal Transport for Kernel Gaussian Mixture Models](https://mlsanity.com/paper/2310.18586/) (2023)
- [On Barycenter Computation: Semi-Unbalanced Optimal Transport-based Method on Gaussians](https://mlsanity.com/paper/2410.08117/) (2024)

## Abstract (arXiv)

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.