Metric Geometry of Spaces of Persistence Diagrams

Abstract

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.

Basic Metric Geometry of the Bottleneck Distance

“…In this section, we collect preliminary material that we will use in the rest of the article. The contents of this section are based on [6], where the reader may find further details. We refer the reader to [5] for basic results on the geometry of metric spaces.…”

“…Several authors have studied the topological and geometric properties of the spaces defined by the bottleneck distance, D ∞ (R 2 ≥0 , ∆), and the p-Wasserstein metrics, D p (R 2 ≥0 , ∆), where R 2 ≥0 = {(x, y) ∈ R 2 : 0 ≤ x ≤ y}, ∆ = {(x, y) ∈ R 2 : x = y}, and 1 ≤ p < ∞, as well as those of generalizations of these spaces [1,2,6,9]. In [6], the authors study the geometry and topology of spaces of generalized persistence diagrams D p (X, A), 1 ≤ p < ∞. These spaces are generalization of the spaces defined in [10, Definition 4].…”

Topological and metric properties of spaces of generalized persistence diagrams

“…Che, Galaz-García, Guijarro, and Mebrillo Solis [15] were the first to study the metric properties of formal sums on metric pairs. They consider metric spaces (X, d) in which d(x, y) < ∞ for all x, y ∈ X and for which d(x, y) > 0 if x = y and p ∈ [1, ∞).…”