Fri, 07 Mar 2025
15:00

L4

Central limit theorems and the smoothed bootstrap in topological data analysis

Johannes Krebs

(Katholische Universitat Eichstätt-Ingolstadt)

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Abstract

We study central limit theorems for persistent Betti numbers and the Euler characteristic of random simplicial complexes built from Poisson and Binomial processes in the critical regime. The approach relies on the idea of stabilizing functionals and dates back to Kesten and Lee (1996) as well as Penrose and Yukich (2001).

However, in many situations such limit theorems prove difficult to use in practice, motivating the use of a bootstrap approach, a resampling technique in mathematical statistics. To this end, we investigate multivariate bootstrap procedures for general stabilizing statistics with a specific focus on the application to topological data analysis. We show that a smoothed bootstrap procedure gives a consistent estimation. Specific statistics considered for the bootstrap include persistent Betti numbers and Euler characteristics of Čech and Vietoris-Rips complexes.

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