Universal Distances for Extended Persistence

TL;DR

This work establishes that the bottleneck distance on extended persistence diagrams is universal for stable comparisons of PL functions. By encoding all relative interlevel set data into a strip-shaped diagram h(f):M→Vect_K and defining the extended diagram Dgm(f) via a functor on M, the authors prove a realizability theorem: every finite stable pairing of diagrams can be realized by PL functions with distance equal to the bottleneck distance. They introduce a lifting framework on M to construct point- and multiset-lifts, enabling explicit realizations and geodesics in the diagram space. The results clarify the relationship between extended persistence, subdiagrams, and interleaving concepts, and they demonstrate that interleaving distance for sheaves (and Reeb graphs) is not intrinsic, highlighting fundamental differences between these canonical invariants and their metrics.

Abstract

The extended persistence diagram is an invariant of piecewise linear functions, which is known to be stable under perturbations of functions with respect to the bottleneck distance as introduced by Cohen-Steiner, Edelsbrunner, and Harer. We address the question of universality, which asks for the largest possible stable distance on extended persistence diagrams, showing that a more discriminative variant of the bottleneck distance is universal. Our result applies more generally to settings where persistence diagrams are considered only up to a certain degree. We achieve our results by establishing a functorial construction and several characteristic properties of relative interlevel set homology, which mirror the classical Eilenberg--Steenrod axioms. Finally, we contrast the bottleneck distance with the interleaving distance of sheaves on the real line by showing that the latter is not intrinsic, let alone universal. This particular result has the further implication that the interleaving distance of Reeb graphs is not intrinsic either.

Figures (19)

• Figure 1.1: Regions in the strip $\mathbb{M}$ corresponding to the ordinary, relative, and extended subdiagrams Cohen-Steiner2009. The support of extended persistence diagrams is shaded in grey.
• Figure 2.1: The strip $\mathbb{M}$ and the image of the embedding $\blacktriangle \colon \overline{\mathbb{R}} \rightarrow \mathbb{M}$.
• Figure 2.2: Incidences defining $T$.
• Figure 2.3: The fundametal domain $\textcolor{DimGrey}{D} := \textcolor{Peru}{\uparrow \operatorname{Im} \blacktriangle} \setminus \textcolor{DarkCyan}{T(\uparrow \operatorname{Im} \blacktriangle)}$.
• Figure 2.4: The tessellation of $\mathbb{M}$ induced by $T$ and $D$.

Theorems & Definitions (71)

• Theorem 1.1
• Proposition 2.1
• Definition 2.2: Extended Persistence Diagram
• Definition 2.3
• Definition 2.4: Matchings relative to the boundary and the Bottleneck Distance
• Remark 2.5
• Lemma 3.1: Homotopy Invariance
• proof
• Lemma 3.1: Mayer--Vietoris Sequence
• proof