Generalized Persistence Diagrams for Persistence Modules over Posets

TL;DR

This paper introduces a generalized framework for persistence diagrams of persistence modules over posets, extending existing concepts to broader categories and settings, including set-theoretic approaches and Reeb graphs.

Contribution

It generalizes the notion of rank invariant and persistence diagrams to arbitrary categories and posets, enabling analysis beyond interval decomposable modules and vector spaces.

Findings

Generalized persistence diagrams can be defined for any poset-indexed functor.

The barcode of Reeb graphs can be obtained without vector space structures.

Lipschitz continuity of persistence diagrams is established in a set-theoretic context.

Abstract

When a category $\mathcal{C}$ satisfies certain conditions, we define the notion of rank invariant for arbitrary poset-indexed functors $F:\mathbf{P} \rightarrow \mathcal{C}$ from a category theory perspective. This generalizes the standard notion of rank invariant as well as Patel's recent extension. Specifically, the barcode of any interval decomposable persistence modules $F:\mathbf{P} \rightarrow \mathbf{vec}$ of finite dimensional vector spaces can be extracted from the rank invariant by the principle of inclusion-exclusion. Generalizing this idea allows freedom of choosing the indexing poset $\mathbf{P}$ of $F: \mathbf{P} \rightarrow \mathcal{C}$ in defining Patel's generalized persistence diagram of $F$. Of particular importance is the fact that the generalized persistence diagram of $F$ is defined regardless of whether $F$ is interval decomposable or not. By specializing our idea to zigzag persistence modules, we also show that the barcode of a Reeb graph can be obtained in a purely set-theoretic setting without passing to the category of vector spaces. This leads to a promotion of Patel's semicontinuity theorem about type $\mathcal{A}$ persistence diagram to Lipschitz continuity theorem for the category of sets.