The Annals of Applied Probability

Limit theorems for Betti numbers of extreme sample clouds with application to persistence barcodes

Takashi Owada

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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.

Article information

Source
Ann. Appl. Probab., Volume 28, Number 5 (2018), 2814-2854.

Dates
Received: April 2017
First available in Project Euclid: 28 August 2018

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1535443235

Digital Object Identifier
doi:10.1214/17-AAP1375

Mathematical Reviews number (MathSciNet)
MR3847974

Zentralblatt MATH identifier
06974766

Subjects
Primary: 60G70: Extreme value theory; extremal processes 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 55U10: Simplicial sets and complexes

Keywords
Extreme value theory random topology persistent homology Betti number central limit theorem Poisson limit theorem

Citation

Owada, Takashi. Limit theorems for Betti numbers of extreme sample clouds with application to persistence barcodes. Ann. Appl. Probab. 28 (2018), no. 5, 2814--2854. doi:10.1214/17-AAP1375. https://projecteuclid.org/euclid.aoap/1535443235


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