Interleaving Distance as a Limit

TL;DR

This paper extends the theoretical framework of interleaving distances in persistent homology, demonstrating how to compare data representations via algebraic structures and limits, thereby generalizing and refining previous isometry theorems.

Contribution

It introduces a new approach to matchings in interleaving theory, using an enlarged category to establish an isometry theorem for more general metrics and data-derived posets.

Findings

Matching theorem fails for arbitrary weights but can be restored with shifted matchings.

An extended category allows for an isometry theorem in a more general setting.

Classical interleaving distance is recovered as a limit of generalized algebraic comparisons.

Abstract

Persistent homology is a way of determining the topological properties of a data set. It is well known that each persistence module admits the structure of a representation of a finite totally ordered set. In previous work, the authors proved an analogue of the isometry theorem of Bauer and Lesnick for representations of a certain class of finite posets. The isometry was between the interleaving metric of Bubenik, de Silva and Scott and a bottleneck metric which incorporated algebraic information. The key step in both isometry theorems was proving a matching theorem, that an interleaving gives rise to a matching of the same height. In this paper we continue this work, restricting to those posets which arise from data while making more general the choice of metrics. We first show that while an interleaving always produces a matching, for an arbitrary choice of weights it will not produce one of the same height. We then show that although the matching theorem fails in this sense, one obtains a "shifted" matching (of the correct height) from an interleaving by enlarging the category. We then prove an isometry theorem on this extended category. As an application, we make precise the way in which representations of finite partially ordered sets approximate persistence modules. Specifically, given two finite point clouds of data, we associate a generalized sequence (net) of algebras over which the persistence modules for both data sets can be compared. We recover the classical interleaving distance uniformly by taking limits.