The reflection distance between zigzag persistence modules

TL;DR

This paper introduces the reflection distance, a new metric for zigzag modules, which bounds the $\, ext{l}^1$-bottleneck distance between their persistence diagrams, enhancing the understanding of zigzag persistence.

Contribution

The paper defines the reflection distance on zigzag modules using reflection functors, providing a new way to compare persistence modules and relate it to existing diagram distances.

Findings

Reflection distance bounds the $\, ext{l}^1$-bottleneck distance between persistence diagrams.

The reflection distance is a metric on zigzag modules of fixed length.

The approach connects reflection functors with stability in persistence modules.

Abstract

By invoking the reflection functors introduced by Bernstein, Gelfand, and Ponomarev in 1973, in this paper we define a metric on the space of all zigzag modules of a given length, which we call the reflection distance. We show that the reflection distance between two given zigzag modules of the same length is an upper bound for the $\ell^1$-bottleneck distance between their respective persistence diagrams.