Theo Lacombe

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  Spaces of persistence diagrams and measures, an optimal transport viewpoint
  Theo Lacombe
  Persistence diagrams (PD) are routinely used in topological data analysis as descriptors to encode the topological properties of some object. These diagrams can be compared with a partial matching metric, sometimes called the "Wasserstein distance between persistence diagrams" due to its important similarities with the metrics used in optimal transport literature, although an explicit connection between these two formalisms was yet to come. By considering the space of persistence diagrams as a measure space, we reformulate its metrics as optimal partial transport problems and introduce a generalization of persistence diagrams, namely Radon measures supported on the upper half plane. Such measures naturally appear in topological data analysis when considering continuous representations of persistence diagrams (e.g. persistence surfaces) but also as expectations of probability distributions on the space of persistence diagrams. We will showcase the strength of this optimal-transport-based formalism on two problems arising in topological data analysis. First, we provide a characterization of convergence in the space of persistence diagrams (with respect to their standard metrics) in terms of vague convergence of measures. This result provides a powerful tool to study continuity properties in this space; in particular it gives an exhaustive characterization of continuous linear representations of persistence diagrams, a common tool used when incorporating persistence diagrams in machine learning pipelines. Second, this formalism allows us to prove new results regarding persistence diagrams in a random setting, as it enables to manipulate some limit objects such as expected persistence diagrams (that are not standard persistence diagrams) and to prove convergence rates and stability results in this context.