Generalized Persistence Diagrams for Persistence Modules over Posets

Abstract

When a category $\mathcal{C}$ satisfies certain conditions, we define the notion of rank invariant for arbitrary poset-indexed functors $F:\mathbf{P}\to \mathcal{C}$ 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 $F:\mathbf{P}\to \mathbf{vec}$ 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 $F:\mathbf{P}\to \mathcal{C}$ 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.

The current version was accepted the Journal of Applied and Computational Topology, except that it contains a new appendix, Appendix H, where we establish the stability of our generalized rank invariant in a suitable setting. Version 4 contains alternative proofs of Proposition 3.17 under the assumption that the indexing poset P is the zigzag poset ZZ (in relation to Botnan and Lesnick's work)

References in corpus (1)

• Graded persistence diagrams and persistence landscapes

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