Metric geometry of spaces of persistence diagrams

Mauricio Che, Fernando Galaz-García, Luis Guijarro, Ingrid Amaranta Membrillo Solis

Abstract: 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 \({\mathcal {D}}_p\), \(1\le p \le \infty \), 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 \({\mathcal {D}}_p(X,A)\), \(1\le p \le \infty \), 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, \({\mathcal {D}}_{{p}}({\mathbb {R}}^{2n},\Delta _n)\), \(1\le n\) and \(1\le p<\infty \), has infinite covering, Hausdorff, asymptotic, Assouad, and Assouad–Nagata dimensions.