This short paper presents a general approach for computing robust Wasserstein barycenters of persistence diagrams. The classical method consists in computing assignment arithmetic means after finding the optimal transport plans between the barycenter and the persistence diagrams. However, this procedure only works for the transportation cost related to the $q$-Wasserstein distance $W_q$ when $q=2$. We adapt an alternative fixed-point method to compute a barycenter diagram for generic transportation costs ($q > 1$), in particular those robust to outliers, $q \in (1,2)$. We show the utility of our work in two applications: \emph{(i)} the clustering of persistence diagrams on their metric space and \emph{(ii)} the dictionary encoding of persistence diagrams. In both scenarios, we demonstrate the added robustness to outliers provided by our generalized framework. Our Python implementation is available at this address: https://github.com/Keanu-Sisouk/RobustBarycenter .

翻译：本文提出了一种计算持久性图表鲁棒Wasserstein重心的通用方法。经典方法是在找到重心与持久性图表之间的最优传输方案后，计算分配算术平均值。然而，该过程仅适用于与$q$-Wasserstein距离$W_q$相关的传输成本，且仅限于$q=2$的情况。我们采用了一种替代的定点方法来计算适用于一般传输成本（$q > 1$）的重心图表，特别是对异常值具有鲁棒性的$q \in (1,2)$情形。我们通过两个应用展示了本研究的实用性：\emph{(i)} 在度量空间中对持久性图表进行聚类，以及\emph{(ii)} 持久性图表的字典编码。在这两种场景中，我们证明了广义框架所提供的增强异常值鲁棒性。我们的Python实现可通过以下地址获取：https://github.com/Keanu-Sisouk/RobustBarycenter。