On the complexity of zero-dimensional multiparameter persistence

TL;DR

This paper investigates the complexity of zero-dimensional multiparameter persistence modules, revealing that even restricted classes contain infinitely many indecomposables, highlighting inherent theoretical challenges.

Contribution

It characterizes finitely generated zero-dimensional multiparameter persistence modules and identifies a class of decomposable representations, exposing their infinite diversity.

Findings

Identified a natural class of decomposable modules.

Proved the existence of infinitely many indecomposables.

Characterized modules under certain assumptions.

Abstract

Multiparameter persistence is a natural extension of the well-known persistent homology, which has attracted a lot of interest. However, there are major theoretical obstacles preventing the full development of this promising theory. In this paper we consider the interesting special case of multiparameter persistence in zero dimensions which can be regarded as a form of multiparameter clustering. In particular, we consider the multiparameter persistence modules of the zero-dimensional homology of filtered topological spaces when they are finitely generated. Under certain assumptions, we characterize such modules and study their decompositions. In particular we identify a natural class of representations that decompose and can be extended back to form zero-dimensional multiparameter persistence modules. Our study of this set of representations concludes that despite the restrictions, there are still infinitely many classes of indecomposables in this set.