Algebraic stability theorem for derived categories of zigzag persistence modules

TL;DR

This paper establishes an algebraic stability theorem for zigzag persistence modules using derived categories and compares it with existing distances, advancing the theoretical understanding of zigzag persistence in topological data analysis.

Contribution

It introduces a new algebraic stability theorem for zigzag persistence modules via derived category methods and relates it to existing distances in the field.

Findings

Derived equivalence between ordinary and zigzag persistence modules.

Defined and computed distances on the derived category of zigzag modules.

Proved an algebraic stability theorem for these distances.

Abstract

We study distances on zigzag persistence modules from the viewpoint of derived categories and Auslander--Reiten quivers. The derived category of ordinary persistence modules is derived equivalent to that of arbitrary zigzag persistence modules, depending on a classical tilting module. Through this derived equivalence, we define and compute distances on the derived category of arbitrary zigzag persistence modules and prove an algebraic stability theorem. We also compare our distance with the distance for purely zigzag persistence modules introduced by Botnan--Lesnick and the sheaf-theoretic convolution distance due to Kashiwara--Schapira.