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 \(\mathcal{D}_p\), \(1\leq p \leq\infty\), that assign, to each metric pair \((X,A)\), a pointed metric space \(\mathcal{D}_p(X,A)\). Moreover, we show that \(\mathcal{D}_{\infty}\) is sequentially continuous with respect to the Gromov-Hausdorff convergence of metric pairs, and we prove that \(\mathcal{D}_p\) preserves several useful metric properties, such as completeness and separability, for \(p \in [1,\infty)\), and geodesicity and non-negative curvature in the sense of Alexandrov, for \(p=2\). 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 \(\mathcal{D}_p(X,A)\), \(1\leq p \leq\infty\), 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, \(\mathcal{D}_{p}(\mathbb{R}^{2n},\Delta_n)\), \(1\leq n\) and \(1\leq p