Metric geometry of spaces of persistence diagrams

Persistence diagrams are objects that play a central role in topological data analysis. In the present article, we investigate the local and global geometric properties of spaces of persistence diagrams. In order to do this, we construct a family of functors $$ , $$ , that assign, to each metric pair (X, A), a pointed metric space $$ . Moreover, we show that $$ is sequentially continuous with respect to the Gromov-Hausdorff convergence of metric pairs, and we prove that $$ preserves several useful metric properties, such as completeness and separability, for $$ , and geodesicity and non-negative curvature in the sense of Alexandrov, for $$ . For the latter case, we describe the metric of the space of directions at the empty diagram. We also show that the Fréchet mean set of a Borel probability measure on $$ , $$ , with finite second moment and compact support is non-empty. As an application of our geometric framework, we prove that the space of Euclidean persistence diagrams, $$ , $$ and $$ , has infinite covering, Hausdorff, asymptotic, Assouad, and Assouad-Nagata dimensions.