On the Hausdorff stability of barcodes over posets

TL;DR

This paper proves a Lipschitz stability result for the Hausdorff distance between barcodes of interval-decomposable modules over general posets, extending stability results beyond the classical one-parameter case.

Contribution

It establishes a general stability theorem for Hausdorff distance in the context of interval-decomposable modules over arbitrary posets, with new tools for morphisms and interleavings.

Findings

Hausdorff distance is Lipschitz stable under interleaving distance.

Develops geometric characterizations of morphisms between interval modules.

Extends stability results to modules over arbitrary posets.

Abstract

The Isometry Theorem of Chazal et al. and Lesnick is a fundamental result in persistence theory, which states that the interleaving distance between two one-parameter persistence modules is equal to the bottleneck distance between their barcodes. Significant effort has been devoted to extending this result to modules defined over more general posets. As these modules do not generally admit nice decompositions, one must restrict attention to the class of interval-decomposable modules in order to define an appropriate notion of bottleneck distance. Even with this assumption, it is known that bottleneck distance may not be equivalent to interleaving distance, but that it is Lipschitz stable under certain, fairly restrictive, assumptions. In this paper, we consider the more basic question of stability of the Hausdorff distance with respect to interleaving distance for interval-decomposable modules. Our main theorem is a Lipschitz stability result, which holds in a fairly general setting of interval-decomposable modules over arbitrary posets, where intervals are assumed to be taken from any family satisfying certain closure conditions. Along the way, we develop some new tools and results for interval-decomposable modules over arbitrary posets, in the form of geometrically-flavored characterizations of the existence of morphisms and interleavings between interval modules.