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 $D_p$ , $1\le p\le \infty$ , that assign, to each metric pair (X, A), a pointed metric space $D_p(X,A)$ . Moreover, we show that $D_{\infty }$ is sequentially continuous with respect to the Gromov-Hausdorff convergence of metric pairs, and we prove that $D_p$ 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 $D_p(X,A)$ , $1\le p\le \infty$ , 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, $D_p(R^{2n},{\Delta }_n)$ , $1\le n$ and $1\le p<\infty$ , has infinite covering, Hausdorff, asymptotic, Assouad, and Assouad-Nagata dimensions.

Keywords: Alexandrov spaces; Asymptotic dimension; Fré chet mean set; Gromov–Hausdorff convergence; Metric pairs; Persistence diagram.