Metric Geometry of Spaces of Persistence Diagrams

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.

Final version. To appear in the Journal of Applied and Computational Topology. 39 pages

References in corpus (1)

• Virtual persistence diagrams, signed measures, Wasserstein distances, and Banach spaces

Cited by in corpus (1)

• Virtual persistence diagrams, signed measures, Wasserstein distances, and Banach spaces