Region-Aware Wasserstein Distances of Persistence Diagrams and Merge Trees

TL;DR

This work extends Wasserstein distances to region-aware persistence diagrams and merge trees by augmenting topological features with vertex-level regional geometry, enabling discriminative comparisons beyond birth/death values. A region-aware ground metric, region-aware projection, and a region-aware Wasserstein distance are defined, with a controllable operator to scale the influence of regions and preserve classical metrics as special cases. The authors prove metric and stability properties, and introduce efficiency strategies via subsampling and compression. Applied to time-varying ensembles, the method enables reliable feature tracking and dimensionality reduction, producing more informative visualizations and clustering indicators. A C++ implementation is provided, demonstrating practical utility across diverse SciVis datasets and establishing a foundation for further geometry-aware topological analysis.

Abstract

This paper presents a generalization of the Wasserstein distance for both persistence diagrams and merge trees [20], [66] that takes advantage of the regions of their topological features in the input domain. Specifically, we redefine the comparison of topological features as a distance between the values of their extrema-aligned regions. It results in a more discriminative metric than the classical Wasserstein distance and generalizes it through an input parameter adjusting the impact of the region properties in the distance. We present two strategies to control both computation time and memory storage of our method by respectively enabling the use of subsets of the regions in the computation, and by compressing the regions' properties to obtain low-memory representations. Extensive experiments on openly available ensemble data demonstrate the efficiency of our method, with running times on the orders of minutes on average. We show the utility of our contributions with two applications. First, we use the assignments between topological features provided by our method to track their evolution in time-varying ensembles and propose the temporal persistence curves to facilitate the understanding of how these features appear, disappear and change over time. Second, our method allows to compute a distance matrix of an ensemble that can be used for dimensionality reduction purposes and visually represent in 2D all its members, we show that such distance matrices also allow to detect key phases in the ensemble. Finally, we provide a C++ implementation that can be used to reproduce our results.

Figures (15)

• Figure 1: Visual analysis of the 48 time steps of the Isabel ensemble (wind velocity) with our novel region-aware Wasserstein distance. Members at different points in time are shown in $(a)$, $(b)$, $(d)$ and $(e)$. The persistence diagram of each member (black insets) allows to identify its features of interest, their maxima are visualized in the input domain with a sphere whose size is proportionnal to their persistence (a notion of importance) and the regions of the highest persistent features are shown with a surface. Our method provides assignments between the features of consecutive members, allowing to track their evolution over time (with matched features having the same color). In this example, it correctly matches the same features together over the whole ensemble while the classical Wasserstein distance $W^{\mathcal{D}}_2$ provide some mismatches (red arrows in red insets of $(a)$ and $(d)$) preventing their correct tracking. The temporal persistence curves in $(c)$ represent the evolution over time of the persistence of the matched features with our method, only curves for the highest persistent features are shown with the same color than in the input domain. This allows to visually convey the changes in the ensemble in one view and permits the user to identify and select important time steps to be visualized. For instance, in the second time window $(b)$ the blue feature starts to appear and grow, while in the last time window $(e)$ the white feature overtake the other ones.
• Figure 2: Illustration of the topological descriptors studied in this work, computed on a 2D minimal example (a) and a noisy version of it (b). For all descriptors, the color code represents the persistence of the corresponding saddle-maximum pair. Critical points are visualized with spheres, the larger ones indicating maxima. Persistence diagrams, merge trees, and branch decomposition trees (BDTs) are shown in the top, center, and bottom insets, respectively. In both datasets, the four primary features (the most prominent hills) are highlighted with salient pairs in the diagram and the merge tree. To maintain clarity in the visualization, pairs with low persistence (below 10% of the function range) are displayed with small white arcs and spheres, while the larger and colored ones correspond to those with higher persistence.
• Figure 3: The persistence diagram, $\mathcal{D}(f_i)$, and the merge tree, $\mathcal{T}(f_i)$, provide a visual summary of the number, data range, and salience of the features of interest present in the data. However, they represent each feature without considering its geometric properties. This results in a lack of specificity for these objects, which can yield identical data representations for significantly distinct datasets. In this example, the four features of each dataset are identically encoded in the corresponding topological representation whereas their respective geometry differs. These datasets can therefore not be discriminated using the celebrated Wasserstein distance for topological representations.
• Figure 4: The Wasserstein distance $W^{\mathcal{T}}_2$ between two BDTs $\mathcal{B}(f_i)$ and $\mathcal{B}(f_j)$ optimizes an assignment between their pairs in the birth/death space (green arrows, right) with a search space constrained by the structure of the trees. It amounts to search for a common contiguous structure between the BDTS (cyan halo). When $\epsilon_1 = 1$, the tree structures do not constrain the search space anymore and $W^{\mathcal{T}}_2$ get back to the Wasserstein distance between persistence diagrams $W^{\mathcal{D}}_2$ (black arrows, right).
• Figure 5: The Wasserstein distance $W^{\mathcal{D}}_2$ matches topological features having overall close birth and death values. In this example, the geometry of the features with black and green spheres are significantly different (circle shape and star shape). For these two datasets, the death value of these features have been swapped resulting in their incorrect matching according $W^{\mathcal{D}}_2$ (red dashes). In contrast, our metric $W^{\mathcal{R}\mathcal{D}}_{2}$ takes into account their geometry through their regions in the input domain and provide a correct matching.