Persistence diagrams are geometric 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 Dp, 1 ≤ p ≤ ∞, that assign, to each metric pair (X, A), a pointed metric space Dp(X, A). Moreover, we show that D∞ is continuous with respect to the Gromov-Hausdorff convergence of metric pairs, and we prove that Dp preserves several useful metric properties, such as completeness and separability, for p ∈ [1, ∞], and geodesicity and non-negative curvature in the sense of Alexandrov, for p = 2. As an application of our framework, we prove that the space of Euclidean persistence diagrams, D2(R 2n , ∆n), has infinite covering, Hausdorff, and asymptotic dimensions.
In this paper, we adapt work of Z.-D. Liu to prove a ball covering property for non-branching $${\mathsf {CD}}$$
CD
spaces with non-negative curvature outside a compact set. As a consequence, we obtain uniform bounds on the number of ends of such spaces.
Given a metric pair (X, A), i.e. a metric space X and a distinguished closed set A ⊂ X, one may construct in a functorial way a pointed pseudometric space D∞(X, A) of persistence diagrams equipped with the bottleneck distance. We investigate the basic metric properties of the spaces D∞(X, A) and obtain characterizations of their metrizability, completeness, separability, and geodesicity.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.