Equivalence of Landscape and Erosion Distances for Persistence Diagrams

TL;DR

This paper proves that erosion and landscape distances for persistence diagrams are equivalent, linking their theoretical and computational aspects, and explores the geometric properties of the erosion distance in topological data analysis.

Contribution

It establishes the equivalence between erosion and landscape distances, connecting their theoretical frameworks and analyzing the geometric properties of erosion distance.

Findings

Erosion and landscape distances are equal.

Erosion distance is not a length metric.

Erosion distance does not embed into any Hilbert space.

Abstract

This paper establishes connections between three of the most prominent metrics used in the analysis of persistence diagrams in topological data analysis: the bottleneck distance, Patel's erosion distance, and Bubenik's landscape distance. Our main result shows that the erosion and landscape distances are equal, thereby bridging the former's natural category-theoretic interpretation with the latter's computationally convenient structure. The proof utilizes the category with a flow framework of de Silva et al., and leads to additional insights into the structure of persistence landscapes. Our equivalence result is applied to prove several results on the geometry of the erosion distance. We show that the erosion distance is not a length metric, and that its intrinsic metric is the bottleneck distance. We also show that the erosion distance does not coarsely embed into any Hilbert space, even when restricted to persistence diagrams arising from degree-0 persistent homology. Moreover, we show that erosion distance agrees with bottleneck distance on this subspace, so that our non-embeddability theorem generalizes several results in the recent literature.