An Isometry Theorem for Generalized Persistence Modules

TL;DR

This paper extends the isometry theorem to a broad class of generalized persistence modules, providing a metric framework for analyzing topological features in data across various posets.

Contribution

It proves an isometry theorem for the interleaving metric on a wide class of generalized persistence modules, generalizing previous results for one-dimensional cases.

Findings

The interleaving metric induces a meaningful metric space structure.

The theorem applies to a large class of posets.

It facilitates distinguishing noise from genuine topological features.

Abstract

In recent work, generalized persistence modules have proved useful in distinguishing noise from the legitimate topological features of a data set. Algebraically, generalized persistence modules can be viewed as representations for the poset algebra. The interplay between various metrics on persistence modules has been of wide interest, most notably, the isometry theorem of Bauer and Lesnick for (one-dimensional) persistence modules. The interleaving metric of Bubenik, de Silva and Scott endows the collection of representations of a poset with values in any category with the structure of a metric space. This metric makes sense for any poset, and has the advantage that post-composition by any functor is a contraction. In this paper, we prove an isometry theorem using this interleaving metric on a full subcategory of generalized persistence modules for a large class of posets.