Generalized Persistence Diagrams for Persistence Modules over Posets
Woojin Kim, Facundo Memoli

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 satisfies certain conditions, we define the notion of rank invariant for arbitrary poset-indexed functors 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 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 of in defining Patel's generalized persistence diagram of . Of particular importance is the fact that the generalized persistence diagram of is defined regardless of whether is interval decomposable or not. By specializing our…
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