The Whole in the Parts: Putting $n$D Persistence Modules Inside Indecomposable $(n + 1)$D Ones

TL;DR

This paper explores the algebraic structure of multidimensional persistence modules, demonstrating how lower-dimensional modules can be embedded into higher-dimensional indecomposable modules, with new constructions that extend previous results.

Contribution

It extends prior work by removing the rectangle-decomposability requirement and constructs indecomposable higher-dimensional modules containing all lower-dimensional modules.

Findings

Any finite rectangle-decomposable nD module is a hyperplane restriction of an (n+1)D indecomposable module.

Constructs an indecomposable (n+1)D module containing all nD modules as hyperplane restrictions over countable fields.

Provides a minimal, improved construction for the case n=1.

Abstract

Multidimensional persistence has been proposed to study the persistence of topological features in data indexed by multiple parameters. In this work, we further explore its algebraic complications from the point of view of higher dimensional indecomposable persistence modules containing lower dimensional ones as hyperplane restrictions. Our previous work constructively showed that any finite rectangle-decomposable $n$D persistence module is the hyperplane restriction of some indecomposable $(n+1)$D persistence module, as a corollary of the result for $n=1$. Here, we extend this by dropping the requirement of rectangle-decomposability. Furthermore, in the case that the underlying field is countable, we construct an indecomposable $(n+1)$D persistence module containing all $n$D persistence modules, up to isomorphism, as hyperplane restrictions. Finally, in the case $n=1$, we present a minimal construction that improves our previous construction.