This short paper presents a general approach for computing robust Wasserstein barycenters (Agueh et al. 2011), (Turner et al. 2014),(Vidal et al. 2020) 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 $q$q-Wasserstein distance $W_{q}$Wq when $q=2$q=2. We adapt an alternative fixed-point method (Tanguy et al. 2025) to compute a barycenter diagram for generic transportation costs ($q > 1$q>1), in particular those robust to outliers, $q \in (1,2)$q∈(1,2). We show the utility of our work in two applications: (i) the clustering of persistence diagrams on their metric space and (ii) the dictionary encoding of persistence diagrams (Sisouk et al. 2024). In both scenarios, we demonstrate the added robustness to outliers provided by our generalized framework.

This article presents a computational framework for the concise encoding of an ensemble of persistence diagrams, in the form of weighted Wasserstein barycenters Turner et al. (2014), Vidal et al. (2020) of a dictionary of atom diagrams. We introduce a multi-scale gradient descent approach for the efficient resolution of the corresponding minimization problem, which interleaves the optimization of the barycenter weights with the optimization of the atom diagrams. Our approach leverages the analytic expressions for the gradient of both sub-problems to ensure fast iterations and it additionally exploits shared-memory parallelism. Extensive experiments on public ensembles demonstrate the efficiency of our approach, with Wasserstein dictionary computations in the orders of minutes for the largest examples. We show the utility of our contributions in two applications. First, we apply Wassserstein dictionaries to data reduction and reliably compress persistence diagrams by concisely representing them with their weights in the dictionary. Second, we present a dimensionality reduction framework based on a Wasserstein dictionary defined with a small number of atoms (typically three) and encode the dictionary as a low dimensional simplex embedded in a visual space (typically in 2D). In both applications, quantitative experiments assess the relevance of our framework. Finally, we provide a C++ implementation that can be used to reproduce our results.