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On approximation theorems for the Euler characteristic with applications to the bootstrap. (English) Zbl 1491.60049

The authors study approximation theorems for the Euler characteristic of the Vietrois-Rips and Čech filtrations obtained from a Poisson or binomial sampling scheme in the critical regime. The main results concern to functional central limits theorems for the Euler characteristic curve associated to such filtrations, the results are applied to the smooth bootstrap of the Euler characteristic where the rate of convergence is determined relative to the Kantorovich-Wasserstein and Kolmogorov metrics. Computer simulations are also provided.
The paper is well written and recommended to those researchers interested in topological data analysis from its statistical point of view.
Let \(X_n\) be a random process that generates point clouds of \(n\) points in \([0,T]^d\). Over these point clouds we can consider the Vietoris-Rips or the Čech filtrations, denoted by \(\mathcal{K}_t(X_n)\), \(t\in[0,T]\). For each \(t\) we can consider the mean Euler characteristic of these filtrations, that define the so called mean Euler characteristic curve. This curve depends only in the underlying distribution of \(X_n\) and the main problem to solve is to provide good estimations for this curve depending on \(n\).
The functional central limit theorem presented in this work is the first to consider the Čech complex, in this sense it is an extension of [A. M. Thomas and T. Owada, Adv. Appl. Probab. 53, No. 1, 57–80 (2021; Zbl 1493.60065)]. Moreover, the application to the binomial case is also new and it is achieved by means of an approximation by Possion schemes. Remark that central limit theorems for persistent Betti numbers obtained from a stationary Poisson process can be found in [Y. Hiraoka et al., Ann. Appl. Probab. 28, No. 5, 2740–2780 (2018; Zbl 1402.60059)].
The authors use a bootstrap technique to estimate the Euler characteristic curve: Given an iid generated cloud point of size \(n\) relative to an unknown density, the authors use a density estimate to replicate the cloud point and compute a bootstrapped Euler characteristic curve from these replicates. The authors show precise estimations for the Kantorovich-Wasserstein and Kolmogorov distances between the bootstrapped and the true curves depending on \(n\) and the supremum distance of the densities. The use of bootstrap for estimation of persistent invariants is not new, see for instance [F. Chazal et al., J. Mach. Learn. Res. 18, Paper No. 159, 40 p. (2018; Zbl 1435.62452)].

MSC:

60F17 Functional limit theorems; invariance principles
62F40 Bootstrap, jackknife and other resampling methods
60B10 Convergence of probability measures
60D05 Geometric probability and stochastic geometry
57R20 Characteristic classes and numbers in differential topology

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