Doubly Regularized Entropic Wasserstein Barycenter

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 (λ,τ)-barycenter, where λ is the inner regularization strength and τ the outer one. This formulation recovers several previously proposed EOT barycenters for various choices of λ,τ≥0 and generalizes them. First, we show that, as 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 τ∼λ2. We discuss also this phenomenon for isotropic Gaussian distributions where all (λ,τ)-barycenters have closed-form. Second, we show that for 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. 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 (λ,τ)-barycenter.