. The Persistence diagrams of three members (a-c) of the Isabel ensemble (wind velocity) concisely and visually encode the number, data range and salience of the features of interest found in the data (eyewall and region of high speed wind, blue and red in (a)). In these diagrams, features with a persistence smaller than 10% of the function range or on the boundary are shown in transparent white. The pointwise mean for these three members (d) exhibits three salient interior features (due to distinct eyewall locations, blue, green and red), although the diagrams of the input members only report two salient interior features at most, located at drastically different data ranges (the red feature is further down the diagonal in (a) and (b)). The Wasserstein barycenter of these three diagrams (e) provides a more representative view of the features found in this ensemble, as it reports a feature number, range and salience that better matches the input diagrams (a-c). Our work introduces a progressive approximation algorithm for such barycenters, with fast practical convergence. Our framework supports computation time constraints (e) which enables the approximation of Wasserstein barycenters within interactive times. We present an application to the clustering of ensemble members based on their persistence diagrams ((f), lifting: α = 0.2), which enables the visual exploration of the main trends of features of interest found in the ensemble. Abstract-This paper presents an efficient algorithm for the progressive approximation of Wasserstein barycenters of persistence diagrams, with applications to the visual analysis of ensemble data. Given a set of scalar fields, our approach enables the computation of a persistence diagram which is representative of the set, and which visually conveys the number, data ranges and saliences of the main features of interest found in the set. Such representative diagrams are obtained by computing explicitly the discrete Wasserstein barycenter of the set of persistence diagrams, a notoriously computationally intensive task. In particular, we revisit efficient algorithms for Wasserstein distance approximation [12, 51] to extend previous work on barycenter estimation [94] . We present a new fast algorithm, which progressively approximates the barycenter by iteratively increasing the computation accuracy as well as the number of persistent features in the output diagram. Such a progressivity drastically improves convergence in practice and allows to design an interruptible algorithm, capable of respecting computation time constraints. This enables the approximation of Wasserstein barycenters within interactive times. We present an application to ensemble clustering where we revisit the k-means algorithm to exploit our barycenters and compute, within execution time constraints, meaningful clusters of ensemble data along with their barycenter diagram. Extensive experiments on synthetic and real-life data sets report that our algorithm converges to barycenters that are qualitatively meaningful with regard to the applications, and quantitatively comparable to previous techniques, while offering an order of magnitude speedup when run until convergence (without time constraint). Our algorithm can be trivially parallelized to provide additional speedups in practice on standard workstations. We provide a lightweight C++ implementation of our approach that can be used to reproduce our results. In that regard, Topological Data Analysis (TDA) [27] has demonstrated over the last two decades its utility to support interactive visualization tasks [45] . In the applications, it robustly and efficiently captures in a generic way the features of interest in scalar data. Examples of successful applications include turbulent combustion [18, 39, 54] , material sciences [33, 42, 44, 80] , nuclear energy [60], fluid dynamics [50], medical imaging [3], chemistry [14, 36] or astrophysics [85, 88] to name a few. An appealing aspect of TDA is the ease it offers for the translation of domain-specific descriptions of features of interest into topological terms. Moreover, to distinguish noise from features, concepts from Persistent Homology [27, 29] provide importance measures, which are both theoretically well established and meaningful in the applications. Among the existing abstractions, such as contour trees [19] , Reeb graphs [16, 66, 93] , or Morse-Smale complexes [25, 40, 41, 43] , the persistence diagram [29] has been extensively studied. In particular, Manuscript
More relevant to our context of urgent decision making, Vidal et al. REF introduced a method for the progressive clustering of persistence diagrams, supporting computation time constraints.
Progressive Wasserstein Barycenters of Persistence Diagrams
IEEE Transactions on Visualization and Computer Graphics
IEEE Transactions on Visualization and Computer Graphics
Computer Science Medicine