Topological and metric properties of spaces of generalized persistence diagrams

Abstract

Motivated by persistent homology and topological data analysis, we consider formal sums on a metric space with a distinguished subset. These formal sums, which we call persistence diagrams, have a canonical 1-parameter family of metrics called Wasserstein distances. We study the topological and metric properties of these spaces. Some of our results are new even in the case of persistence diagrams on the half-plane. Under mild conditions, no persistence diagram has a compact neighborhood. If the underlying metric space is σ-compact then so is the space of persistence diagrams. However, under mild conditions, the space of persistence diagrams is not hemicompact and the space of functions from this space to a topological space is not metrizable. Spaces of persistence diagrams inherit completeness and separability from the underlying metric space. Some spaces of persistence diagrams inherit being path connected, being a length space, and being a geodesic space, but others do not. We give criteria for a set of persistence diagrams to be totally bounded and relatively compact. We also study the curvature and dimension of spaces of persistence diagrams and their embeddability into a Hilbert space. As an important technical step, which is of independent interest, we give necessary and sufficient conditions for the existence of optimal matchings of persistence diagrams.

Basic Metric Geometry of the Bottleneck Distance

“…These two statements are no longer true for D ∞ (X, A): see Example 6.6. The characterization of completeness of D ∞ (X, A) (see [3,Theorem 6.3 & Proposition 6.10]) does agree with the characterization for D ∞ (X, A) (see Theorem B); however, the proof given in [3] does not seem to be easy to adapt for the space D ∞ (X, A).…”

“…Our results are related to some of the theorems in [3], where the authors study the space D ∞ (X, A) of persistence diagrams σ such that the subdiagram {{x ∈ σ | d(x, A) ≥ δ}} is finite for every δ > 0. We point out that D ∞ (X, A) is a closed subspace of D ∞ (X, A).…”

“…For instance, if X is a metric space then so is D ∞ (X, A) (see the proof of [6, Lemma 2.4(2)]); the analogous statement for D ∞ (X, A), however, is far from being true (see Theorem A). In [3,Theorem 6.2], the authors showed that if X is separable then so is D ∞ (X, A), but this is not necessarily true for D ∞ (X, A) (see Example 5.3). Moreover, if A is distance minimizing (that is, for every x ∈ X there exists a ∈ A such that d(x, a) = d(x, A)) then there always exists an optimal bijection between two diagrams in D ∞ (X, A) [3, Theorem 4.9], and if in addition X is geodesic then so is D ∞ (X, A) [3, Theorem 5.10].…”

“…It is also worth mentioning that in [3] the authors deal with extended pseudometric spaces X (see Definition 2.1) which are not necessarily metric spaces. Our proofs should easily carry over to this more general context.…”