Open Access
August 2020 Convergence of persistence diagrams for topological crackle
Takashi Owada, Omer Bobrowski
Bernoulli 26(3): 2275-2310 (August 2020). DOI: 10.3150/20-BEJ1193


In this paper, we study the persistent homology associated with topological crackle generated by distributions with an unbounded support. Persistent homology is a topological and algebraic structure that tracks the creation and destruction of topological cycles (generalizations of loops or holes) in different dimensions. Topological crackle is a term that refers to topological cycles generated by random points far away from the bulk of other points, when the support is unbounded. We establish weak convergence results for persistence diagrams – a point process representation for persistent homology, where each topological cycle is represented by its $({\mathit{birth},\mathit{death}})$ coordinates. In this work, we treat persistence diagrams as random closed sets, so that the resulting weak convergence is defined in terms of the Fell topology. Using this framework, we show that the limiting persistence diagrams can be divided into two parts. The first part is a deterministic limit containing a densely-growing number of persistence pairs with a shorter lifespan. The second part is a two-dimensional Poisson process, representing persistence pairs with a longer lifespan.


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Takashi Owada. Omer Bobrowski. "Convergence of persistence diagrams for topological crackle." Bernoulli 26 (3) 2275 - 2310, August 2020.


Received: 1 October 2019; Revised: 1 January 2020; Published: August 2020
First available in Project Euclid: 27 April 2020

zbMATH: 07193960
MathSciNet: MR4091109
Digital Object Identifier: 10.3150/20-BEJ1193

Keywords: Extreme value theory , Fell topology , Persistent homology , point process , topological crackle

Rights: Copyright © 2020 Bernoulli Society for Mathematical Statistics and Probability

Vol.26 • No. 3 • August 2020
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