Tracking the variety of interleavings

TL;DR

This paper investigates how the algebraic structure of interleaving varieties between persistence modules evolves with the interleaving parameter, focusing on interval modules to classify their progression and extract information.

Contribution

It classifies all possible progressions of interleaving varieties for interval modules and analyzes the information they encode about the modules.

Findings

Classified all possible variety progressions for interval modules.

Determined what module information is captured by the variety progression.

Connected the evolution of varieties to the interleaving distance.

Abstract

In topological data analysis persistence modules are used to distinguish the legitimate topological features of a finite data set from noise. Interleavings between persistence modules feature prominantly in the analysis. One can show that for $\epsilon$ positive, the collection of $\epsilon$-interleavings between two persistence modules $M$ and $N$ has the structure of an affine variety, Thus, the smallest value of $\epsilon$ corresponding to a nonempty variety is the interleaving distance. With this in mind, it is natural to wonder how this variety changes with the value of $\epsilon$, and what information about $M$ and $N$ can be seen from just the knowledge of their varieties. In this paper, we focus on the special case where $M$ and $N$ are interval modules. In this situation we classify all possible progressions of varieties, and determine what information about $M$ and $N$ is present in the progression.