Doubly Regularized Entropic Wasserstein Barycenter

• Published in: Foundations of computational mathematics
• Main Author: Chizat, Lénaïc
• Format: Journal Article
• Language: English
• Published: 12.08.2025
• ISSN: 1615-3375; 1615-3383
• DOI: 10.1007/s10208-025-09724-8

We study a general formulation of regularized Wasserstein barycenters that enjoy favorable regularity, approximation, stability and (grid-free) optimization properties. This barycenter is defined as the unique probability measure that minimizes the sum of entropic optimal transport (EOT) costs with respect to a family of given probability measures, plus an entropy term. We denote it the $$(\lambda ,\tau )$$ ( λ , τ ) -barycenter, where $$\lambda $$ λ is the inner regularization strength and $$\tau $$ τ the outer one. This formulation recovers several previously proposed EOT barycenters for various choices of $$\lambda ,\tau \ge 0$$ λ , τ ≥ 0 and generalizes them. First, we show that, as $$\lambda , \tau \rightarrow 0$$ λ , τ → 0 , regularizing doubly can decrease the approximation error compared to a single regularization. More specifically, we show that for smooth densities and the quadratic cost, the leading order term of the suboptimality in the (unregularized) Wasserstein barycenter objective cancels when $$\tau \sim \frac{\lambda }{2}$$ τ ∼ λ 2 . We discuss also this phenomenon for isotropic Gaussian distributions where all $$(\lambda ,\tau )$$ ( λ , τ ) -barycenters have closed-form. Second, we show that for $$\lambda ,\tau >0$$ λ , τ > 0 , this barycenter has a smooth density and is strongly stable under perturbation of the marginals. In particular, it can be estimated efficiently: given n samples from each of the probability measures, it converges in relative entropy to the population barycenter at a rate $$n^{-1/2}$$ n - 1 / 2 . Finally, this formulation is amenable to a grid-free optimization algorithm: we propose a simple Noisy Particle Gradient Descent method which, in the mean-field limit, converges globally at an exponential rate to the $$(\lambda ,\tau )$$ ( λ , τ ) -barycenter.