🤖 AI Summary
This paper addresses the modeling and optimization of regularized Wasserstein barycenters. To overcome limitations of existing approaches—particularly concerning regularization design, stability, and mesh-free optimization—we propose the doubly regularized entropy-Wasserstein barycenter, which jointly minimizes the entropy-optimal transport (EOT) cost and dual entropy regularization terms (acting on both the barycenter and marginal couplings). Theoretically, when the regularization parameters satisfy τ = λ/2, the estimation bias is O(λ²), yielding asymptotically unbiased estimation; the barycenter density is smooth, and the solution exhibits strong stability against perturbations in marginal distributions. Algorithmically, we employ noisy-particle gradient descent, achieving global exponential convergence in the mean-field limit; with n samples, the relative entropy converges at rate O(n⁻¹/²). Our framework unifies and generalizes major barycenter models, combining theoretical rigor with computational scalability.
📝 Abstract
We study a general formulation of regularized Wasserstein barycenters that enjoys 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 $(lambda, au)$-barycenter, where $lambda$ is the inner regularization strength and $ au$ the outer one. This formulation recovers several previously proposed EOT barycenters for various choices of $lambda, au geq 0$ and generalizes them. First, in spite of -- and in fact owing to -- being emph{doubly} regularized, we show that our formulation is debiased for $ au=lambda/2$: the suboptimality in the (unregularized) Wasserstein barycenter objective is, for smooth densities, of the order of the strength $lambda^2$ of entropic regularization, instead of $max{lambda, au}$ in general. We discuss this phenomenon for isotropic Gaussians where all $(lambda, au)$-barycenters have closed form. Second, we show that for $lambda, au>0$, 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 $n^{-1/2}$. And finally, this formulation lends itself naturally to a grid-free optimization algorithm: we propose a simple emph{noisy particle gradient descent} which, in the mean-field limit, converges globally at an exponential rate to the barycenter.
