August 2023 Bootstrapping persistent Betti numbers and other stabilizing statistics
Benjamin Roycraft, Johannes Krebs, Wolfgang Polonik
Author Affiliations +
Ann. Statist. 51(4): 1484-1509 (August 2023). DOI: 10.1214/23-AOS2277

Abstract

We investigate multivariate bootstrap procedures for general stabilizing statistics, with specific application to topological data analysis. The work relates to other general results in the area of stabilizing statistics, including central limit theorems for geometric and topological functionals of Poisson and binomial processes in the critical regime, where limit theorems prove difficult to use in practice, motivating the use of a bootstrap approach. A smoothed bootstrap procedure is shown to give consistent estimation in these settings. Specific statistics considered include the persistent Betti numbers of Čech and Vietoris–Rips complexes over point sets in Rd, along with Euler characteristics, and the total edge length of the k-nearest neighbor graph. Special emphasis is given to weakening the necessary conditions needed to establish bootstrap consistency. In particular, the assumption of a continuous underlying density is not required. Numerical studies illustrate the performance of the proposed method.

Funding Statement

Benjamin Roycraft was partially supported by the National Science Foundation (NSF), grant number DMS-1148643. Johannes Krebs was partially supported by the German Research Foundation (DFG), grant number KR-4977/2-1. Wolfgang Polonik was partially supported by the National Science Foundation (NSF), grant number DMS-2015575.
Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P. Sloan Foundation, the U.S. Department of Energy Office of Science and the Participating Institutions. SDSS-IV acknowledges support and resources from the Center for High-Performance Computing at the University of Utah. The SDSS website is www.sdss.org.

Acknowledgments

Thank you to the Associate Editor, Editor and reviewers for their helpful comments and thorough examination of this work.

Citation

Download Citation

Benjamin Roycraft. Johannes Krebs. Wolfgang Polonik. "Bootstrapping persistent Betti numbers and other stabilizing statistics." Ann. Statist. 51 (4) 1484 - 1509, August 2023. https://doi.org/10.1214/23-AOS2277

Information

Received: 1 March 2021; Revised: 1 March 2023; Published: August 2023
First available in Project Euclid: 19 October 2023

Digital Object Identifier: 10.1214/23-AOS2277

Subjects:
Primary: 62F40
Secondary: 55N31 , 62G05 , 62H10 , 62R40

Keywords: Betti numbers , bootstrap , Euler characteristic , Persistent homology , Random geometric complexes , stabilizing statistics , Stochastic geometry , topological data analysis

Rights: Copyright © 2023 Institute of Mathematical Statistics

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Vol.51 • No. 4 • August 2023
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