A common challenge in aggregating data from multiple sources can be formalized as an \textit{Optimal Transport} (OT) barycenter problem, which seeks to compute the average of probability distributions with respect to OT discrepancies. However, the presence of outliers and noise in the data measures can significantly hinder the performance of traditional statistical methods for estimating OT barycenters. To address this issue, we propose a novel, scalable approach for estimating the \textit{robust} continuous barycenter, leveraging the dual formulation of the \textit{(semi-)unbalanced} OT problem. To the best of our knowledge, this paper is the first attempt to develop an algorithm for robust barycenters under the continuous distribution setup. Our method is framed as a $\min$-$\max$ optimization problem and is adaptable to \textit{general} cost function. We rigorously establish the theoretical underpinnings of the proposed method and demonstrate its robustness to outliers and class imbalance through a number of illustrative experiments.

本文解决了聚合来自多个来源的数据时的鲁棒重心估计问题，尤其针对传统统计方法在数据中存在离群点和噪声时的不足。提出了一种新的可扩展方法，利用半不平衡最优传输问题的对偶形式，对鲁棒连续重心进行估计，并在理论和实验上验证了其在离群点和类别不平衡情况下的鲁棒性。

基于半不平衡神经最优传输的鲁棒重心估计