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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.
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Functional limit theorems for the euler characteristic process in the critical regime
Abstract This study presents functional limit theorems for the Euler characteristic of Vietoris–Rips complexes. The points are drawn from a nonhomogeneous Poisson process on $$\mathbb{R}^d$$ , and the connectivity radius governing the formation of simplices is taken as a function of the time parameter t , which allows us to treat the Euler characteristic as a stochastic process. The setting in which this takes place is that of the critical regime, in which the simplicial complexes are highly connected and have nontrivial topology. We establish two ‘functional-level’ limit theorems, a strong law of large numbers and a central limit theorem, for the appropriately normalized Euler characteristic process.
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- Award ID(s):
- 1811428
- PAR ID:
- 10233136
- Date Published:
- Journal Name:
- Advances in Applied Probability
- Volume:
- 53
- Issue:
- 1
- ISSN:
- 0001-8678
- Page Range / eLocation ID:
- 57 to 80
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/apr.2020.46
