Realizations of Indecomposable Persistence Modules of Arbitrarily Large Dimension

TL;DR

This paper constructs infinite families of indecomposable persistence modules of arbitrarily large dimension, demonstrating their realizability in topological data analysis and addressing challenges in multidimensional persistence.

Contribution

It introduces an algebraic construction proving the existence of infinite indecomposable modules and shows they can be realized in real data through topological spaces and filtrations.

Findings

Existence of infinite families of indecomposable modules over large grids.

Construction of modules realizable in Vietoris-Rips filtrations.

Indecomposable modules can appear in real data analysis.

Abstract

While persistent homology has taken strides towards becoming a wide-spread tool for data analysis, multidimensional persistence has proven more difficult to apply. One reason is the serious drawback of no longer having a concise and complete descriptor analogous to the persistence diagrams of the former. We propose a simple algebraic construction to illustrate the existence of infinite families of indecomposable persistence modules over regular grids of sufficient size. On top of providing a constructive proof of representation infinite type, we also provide realizations by topological spaces and Vietoris-Rips filtrations, showing that they can actually appear in real data and are not the product of degeneracies.