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Despite the obvious similarities between the metrics used in topological dataanalysis and those of optimal transport, an optimal-transport based formalismto study persistence diagrams and similar topological descriptors has yet tocome. In this article, by considering the space of persistence diagrams as aspace of discrete measures, and by observing that its metrics can be expressedas optimal partial transport problems, we introduce a generalization ofpersistence diagrams, namely Radon measures supported on the upper half plane.Such measures naturally appear in topological data analysis when consideringcontinuous representations of persistence diagrams (e.g.\ persistence surfaces)but also as limits for laws of large numbers on persistence diagrams or asexpectations of probability distributions on the persistence diagrams space. Weexplore topological properties of this new space, which will also hold for theclosed subspace of persistence diagrams. New results include a characterizationof convergence with respect to Wasserstein metrics, a geometric description ofbarycenters (Fr\'echet means) for any distribution of diagrams, and anexhaustive description of continuous linear representations of persistencediagrams. We also showcase the strength of this framework to study randompersistence diagrams by providing several statistical results made meaningfulthanks to this new formalism.

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