The structure and stability of persistence modules

2012/07/16 by Frédéric Chazal, Vin de Silva, Chazal, Frederic +5 · 20 citations
Computer Science · Mathematics · Physics and Astronomy · #Algebraic Topology (math.AT) #Category Theory (math.CT) #Complex Network Analysis Techniques #Computational Geometry (cs.CG) #FOS: Computer and information sciences #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis

paper · doi:10.48550/arxiv.1207.3674

openalex publication_date 2012/07/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We give a self-contained treatment of the theory of persistence modules indexed over the real line. We give new proofs of the standard results. Persistence diagrams are constructed using measure theory. Linear algebra lemmas are simplified using a new notation for calculations on quiver representations. We show that the stringent finiteness conditions required by traditional methods are not necessary to prove the existence and stability of the persistence diagram. We introduce weaker hypotheses for taming persistence modules, which are met in practice and are strong enough for the theory still to work. The constructions and proofs enabled by our framework are, we claim, cleaner and simpler.

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