On approximation theorems for the Euler characteristic with applications to the bootstrap

@article{Krebs2020OnAT,
  title={On approximation theorems for the Euler characteristic with applications to the bootstrap},
  author={Johannes T. N. Krebs and Benjamin Roycraft and Wolfgang Polonik},
  journal={Electronic Journal of Statistics},
  year={2020}
}
We study approximation theorems for the Euler characteristic of the Vietoris-Rips and \v Cech filtration. The filtration is obtained from a Poisson or binomial sampling scheme in the critical regime. We apply our results to the smooth bootstrap of the Euler characteristic and determine its rate of convergence in the Kantorovich-Wasserstein distance and in the Kolmogorov distance. 
View PDF on arXiv
3 Citations
Highly Influential Citations
1
Background Citations
2
Methods Citations
1

Figures and Tables from this paper

3 Citations

Functional strong laws of large numbers for Euler characteristic processes of extreme sample clouds

To recover the topology of a manifold in the presence of heavy tailed or exponentially decaying noise, one must understand the behavior of geometric complexes whose points lie in the tail of these…

Topology-Driven Goodness-of-Fit Tests in Arbitrary Dimensions

This paper adopts a tool from computational topology, the Euler characteristic curve (ECC) of a sample, to perform one- and two-sample goodness of fit tests, we call TopoTests . The presented tests…

A Flexible Approach for Normal Approximation of Geometric and Topological Statistics

  • 2022

References

SHOWING 1-10 OF 47 REFERENCES

Bootstrapping Persistent Betti Numbers and Other Stabilizing Statistics

The present contribution investigates multivariate bootstrap procedures for general stabilizing statistics, with specific application to topological data analysis. Existing limit theorems for…

Central limit theorems for some graphs in computational geometry

Let Bn be an increasing sequence of regions in d-dimensional space with volume n and with union d. We prove a general central limit theorem for functionals of point sets, obtained either by…

On the asymptotic normality of persistent Betti numbers

Persistent Betti numbers are a major tool in persistent homology, a subfield of topological data analysis. Many tools in persistent homology rely on the properties of persistent Betti numbers…

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…

Topology and data

This paper will discuss how geometry and topology can be applied to make useful contributions to the analysis of various kinds of data, particularly high throughput data from microarray or other sources.

New Berry–Esseen bounds for functionals of binomial point processes.

New Berry-Esseen bounds are obtained for set approximations with random tessellations, as well as for functionals of coverage processes, and the classical Hoeffding decompositions are revisited.

The density of expected persistence diagrams and its kernel based estimation

A cross-validation scheme for selecting an optimal bandwidth is proposed, which is proven to be a consistent procedure to estimate the density.

Probability Theory : The Coupling Method

Coupling is a powerful method in probability theory through which random variables can be compared with each other. Coupling has been applied in a broad variety of contexts, e.g. to prove limit…

A NEW METHOD OF NORMAL APPROXIMATION

We introduce a new version of Stein's method that reduces a large class of normal approximation problems to variance bounding exercises, thus making a connection between central limit theorems and…

Persistent homology of the cosmic web – I. Hierarchical topology in ΛCDM cosmologies

Using a set of Lambda cold dark matter simulations of cosmic structure formation, we study the evolving connectivity and changing topological structure of the cosmic web using state-of-the-art…