Indefinite Topological Kernels

Topological Data Analysis (\texttt{TDA}) is a recent and growing branch of statistics devoted to the study of the shape of the data. Motivated by the complexity of the object summarizing the topology of data, we introduce a new topological kernel that allows to extend the \texttt{TDA} toolbox to supervised learning. Exploiting the geodesic structure of the space of Persistence Diagrams, we define a geodesic kernel for Persistence Diagrams, we characterize it, and we show with an application that, despite not being positive semi--definite, it can be successfully used in regression tasks.