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 $(\lambda,\tau)$-barycenter, where $\lambda$ is the inner regularization strength and $\tau$ the outer one. This formulation recovers several previously proposed EOT barycenters for various choices of $\lambda,\tau \geq 0$ and generalizes them. First, in spite of -- and in fact owing to -- being \emph{doubly} regularized, we show that our formulation is debiased for $\tau=\lambda/2$: 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 $\max\{\lambda,\tau\}$ in general. We discuss this phenomenon for isotropic Gaussians where all $(\lambda,\tau)$-barycenters have closed form. Second, we show that for $\lambda,\tau>0$, 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.

翻译：我们研究了一种正则化Wasserstein重心的通用形式，该形式具有良好的正则性、近似性、稳定性和（无网格）优化性质。该重心定义为唯一使得一组给定概率测度上的熵最优传输（EOT）成本之和加上熵项最小化的概率测度。我们将其记为$(\lambda,\tau)$-重心，其中$\lambda$为内部正则化强度，$\tau$为外部正则化强度。该形式针对$\lambda,\tau \geq 0$的不同取值恢复并推广了先前提出的若干EOT重心。首先，尽管——实际上恰恰由于——是双重正则化的，我们证明当$\tau=\lambda/2$时该形式是无偏的：对于光滑密度函数，其在（无正则化的）Wasserstein重心目标中的次优性阶数为熵正则化强度$\lambda^2$，而非通常的$\max\{\lambda,\tau\}$。我们以各向同性高斯分布为例讨论此现象，其中所有$(\lambda,\tau)$-重心均具有闭式解。其次，我们证明当$\lambda,\tau>0$时，该重心具有光滑密度函数，且对边缘分布的扰动具有强稳定性。特别地，它可被高效估计：给定每个概率测度的$n$个样本，其相对熵以$n^{-1/2}$的速率收敛至总体重心。最后，该形式自然适用于无网格优化算法：我们提出一种简单的噪声粒子梯度下降法，在平均场极限下以指数速率全局收敛至重心。