Statistics on topological descriptors based on optimal transport.

Summary Topological data analysis (TDA) extracts rich information from structured data (such as graphs or time series) present in modern learning problems. It will represent this information in the form of descriptors of which persistence diagrams are part, which can be described as point measures supported on a half-plane. Although they are not simple vectors, persistence diagrams can nevertheless be compared with each other using partial matching metrics. The similarity between these metrics and the usual metrics of optimal transport - another field of mathematics - has been known for a long time, but a formal link between these two fields remained to be established. The purpose of this thesis is to clarifier this connection so that we can use the many achievements of optimal transport afin developing new statistical tools (both theoretical and practical) for manipulating persistence diagrams. First, we show how partial optimal transport with boundary, a variant of classical optimal transport, provides us with a formalism that contains the usual DTA metrics. We then illustrate the beneficial contributions of this reformulation in different situations: a theoretical study and algorithm for effictively estimating the barycenters of persistence diagrams using regularized transport, characterization of continuous linear representations of the diagrams and their learning via a versatile neural network, as well as a stability result for linear averages of randomly drawn diagrams.