We study the computation of doubly regularized Wasserstein barycenters, a recently introduced family of entropic barycenters governed by inner and outer regularization strengths. Previous research has demonstrated that various regularization parameter choices unify several notions of entropy-penalized barycenters while also revealing new ones, including a special case of debiased barycenters. In this paper, we propose and analyze an algorithm for computing doubly regularized Wasserstein barycenters. Our procedure builds on damped Sinkhorn iterations followed by exact maximization/minimization steps and guarantees convergence for any choice of regularization parameters. An inexact variant of our algorithm, implementable using approximate Monte Carlo sampling, offers the first non-asymptotic convergence guarantees for approximating Wasserstein barycenters between discrete point clouds in the free-support/grid-free setting.

翻译：我们研究双重正则化Wasserstein重心的计算方法，这是一类最近提出的由内外正则化强度共同控制的熵正则化重心。已有研究表明，不同的正则化参数选择能够统一多种熵惩罚重心的概念，同时揭示新的重心类型，包括去偏重心的特例。本文提出并分析了一种计算双重正则化Wasserstein重心的算法。该算法基于阻尼Sinkhorn迭代，辅以精确的最大化/最小化步骤，并保证对于任意正则化参数选择均能收敛。算法的非精确变体可通过近似蒙特卡洛采样实现，首次在自由支撑/无网格设定下为离散点云之间的Wasserstein重心逼近提供非渐近收敛保证。