# Understanding the Topology and the Geometry of the Persistence Diagram Space via Optimal Partial Transport

We consider a generalization of persistence diagrams, namely Radon measures supported on the upper half plane for which we define natural extensions of Wasserstein and bottleneck distances between persistence diagrams. Such measures naturally appear in topological data analysis when considering continuous representations of persistence diagrams (e.g. persistence surfaces) but also as limits for laws of large numbers on persistence diagrams or as expectations of probability distributions on the persistence diagrams space. Introducing a formalism originating from the theory of optimal partial transport, we build a convenient framework to prove topological properties of this new space, which will also hold for the closed subspace of persistence diagrams. New results include a characterization of convergence with respect to Wasserstein metrics, and the existence of barycenters (Fréchet means) for any distribution of diagrams. We also showcase the strength of this framework by providing several statistical results made meaningful thanks to this new formalism.

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