Metric geometry of spaces of persistence diagrams

Abstract

AbstractPersistence 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 $${\mathcal {D}}_p$$ D p , $$1\le p \le \infty $$ 1 ≤ p ≤ ∞ , that assign, to each metric pair (X, A), a pointed metric space $${\mathcal {D}}_p(X,A)$$ D p ( X , A ) . Moreover, we show that $${\mathcal {D}}_{\infty }$$ D ∞ is sequentially continuous with respect to the Gromov–Hausdorff convergence of metric pairs, and we prove that $${\mathcal {D}}_p$$ D p preserves several useful metric properties, such as completeness and separability, for $$p \in [1,\infty )$$ p ∈ [ 1 , ∞ ) , and geodesicity and non-negative curvature in the sense of Alexandrov, for $$p=2$$ 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 $${\mathcal {D}}_p(X,A)$$ D p ( X , A ) , $$1\le p \le \infty $$ 1 ≤ p ≤ ∞ , 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, $${\mathcal {D}}_{{p}}({\mathbb {R}}^{2n},\Delta _n)$$ D p ( R 2 n , Δ n ) , $$1\le n$$ 1 ≤ n and $$1\le p<\infty $$ 1 ≤ p < ∞ , has infinite covering, Hausdorff, asymptotic, Assouad, and Assouad–Nagata dimensions.