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October 2018 Limit theorems for Betti numbers of extreme sample clouds with application to persistence barcodes
Takashi Owada
Ann. Appl. Probab. 28(5): 2814-2854 (October 2018). DOI: 10.1214/17-AAP1375

Abstract

We investigate the topological dynamics of extreme sample clouds generated by a heavy tail distribution on $\mathbb{R}^{d}$ by establishing various limit theorems for Betti numbers, a basic quantifier of algebraic topology. It then turns out that the growth rate of the Betti numbers and the properties of the limiting processes all depend on the distance of the region of interest from the weak core, that is, the area in which random points are placed sufficiently densely to connect with one another. If the region of interest becomes sufficiently close to the weak core, the limiting process involves a new class of Gaussian processes. We also derive the limit theorems for the sum of bar lengths in the persistence barcode plot, a graphical descriptor of persistent homology.

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Takashi Owada. "Limit theorems for Betti numbers of extreme sample clouds with application to persistence barcodes." Ann. Appl. Probab. 28 (5) 2814 - 2854, October 2018. https://doi.org/10.1214/17-AAP1375

Information

Received: 1 April 2017; Published: October 2018
First available in Project Euclid: 28 August 2018

zbMATH: 06974766
MathSciNet: MR3847974
Digital Object Identifier: 10.1214/17-AAP1375

Subjects:
Primary: 60G70 , 60K35
Secondary: 55U10 , 60D05

Keywords: Betti number , central limit theorem , Extreme value theory , Persistent homology , Poisson limit theorem , random topology

Rights: Copyright © 2018 Institute of Mathematical Statistics

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Vol.28 • No. 5 • October 2018
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