IEEE VIS 2024 Content: Wasserstein Dictionaries of Persistence Diagrams

Wasserstein Dictionaries of Persistence Diagrams

Keanu Sisouk -

Julie Delon -

Julien Tierny -

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DOI: 10.1109/TVCG.2023.3330262

Room: Bayshore I

2024-10-17T14:51:00ZGMT-0600Change your timezone on the schedule page
2024-10-17T14:51:00Z
Exemplar figure, described by caption below
Visual comparison (left) between the input persistence diagrams for three members of an initial ensemble (one member per ground-truth cluster class). For each member, the sphere color encodes the correspondence between the input and the compressed diagrams. This visual comparison shows that the main features of the diagrams are well preserved by our reduction approach, for which a low relative reconstruction error can be observed. The planar overview of the ensemble (right) generated by our dimensionality reduction enables the visualization of the relations between the different diagrams of the ensemble.
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Keywords

Topological data analysis, ensemble data, persistence diagrams

Abstract

This paper presents a computational framework for the concise encoding of an ensemble of persistence diagrams, in the form of weighted Wasserstein barycenters [100], [102] 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.