Generalized Persistence Diagrams

TL;DR

This paper extends the concept of persistence diagrams to a broader setting of constructible persistence modules in symmetric monoidal categories, introducing two types and proving their stability under small perturbations.

Contribution

It generalizes persistence diagrams to new categorical contexts and defines two types with stability guarantees.

Findings

Introduction of type A and type B persistence diagrams.

Proof of stability of both diagrams under small perturbations.

Extension of persistence diagram theory to symmetric monoidal and abelian categories.

Abstract

We generalize the persistence diagram of Cohen-Steiner, Edelsbrunner, and Harer to the setting of constructible persistence modules valued in a symmetric monoidal category. We call this the type A persistence diagram of a persistence module. If the category is also abelian, then we define a second type B persistence diagram. In addition, we show that both diagrams are stable to all sufficiently small perturbations of the module.