Limit theorems for process-level Betti numbers for sparse and critical regimes

Abstract

The objective of this study is to examine the asymptotic behavior of Betti numbers of Čech complexes treated as stochastic processes and formed from random points in the d-dimensional Euclidean space ${\mathbb{R}}^d$ . We consider the case where the points of the Čech complex are generated by a Poisson process with intensity nf for a probability density f. We look at the cases where the behavior of the connectivity radius of the Čech complex causes simplices of dimension greater than $k+1$ to vanish in probability, the so-called sparse regime, as well when the connectivity radius is of the order of $n^{-1/d}$ , the critical regime. We establish limit theorems in the aforementioned regimes: central limit theorems for the sparse and critical regimes, and a Poisson limit theorem for the sparse regime. When the connectivity radius of the Čech complex is $o(n^{-1/d})$ , i.e. the sparse regime, we can decompose the limiting processes into a time-changed Brownian motion or a time-changed homogeneous Poisson process respectively. In the critical regime, the limiting process is a centered Gaussian process but has a much more complicated representation, because the Čech complex becomes highly connected with many topological holes of any dimension.

Testing goodness of fit for point processes via topological data analysis

“…Consider for illustration the term corresponding to the situation x 2 = y 1 , x 1 = y 2 = y 2 , x 3 = y 3 = y 3 . The sum is handled as in the proof of Lemma 9.6(1) by applying (4) and bounding the involved Case Ib: The claim follows by applying the Hlder inequality to (35) and arguing as in Case Ia using Lemma 9.10 to bound the first integral.…”

“…Case II: We apply the Hölder inequality exactly as in (35) and argue as in Case Ia, except that the second integral is first integrated with respect to y 3 rather than y 1 .…”

“…In large domains, this becomes time-consuming. On the other hand, there has been substantial progress towards establishing CLTs in large domains for functionals related to persistent Betti numbers [25,29,35,39,40]. However, these results are restricted to the null hypothesis of complete spatial randomness -i.e., the Poisson point process -and establish asymptotic Gaussianity on a multivariate, but not on a functional level.…”

Strong Law of Large Numbers for Betti Numbers in the Thermodynamic Regime

“…In conclusion, we completely establish the strong law of large numbers in both the Euclidean setting and the manifold setting in this paper. The question on central limit theorem in the Euclidean setting is partially answered in [17,24] under a technical condition that the limiting radius r is small enough. So we can say central limit theorems in both the settings are still open.…”

Harnessing the power of topological data analysis to detect change points

“…Statistical inference for topological summaries within the TDA framework and asymptotic analysis of Betti numbers in particular yet remain largely understudied (see the discussion by Chazal & Michel, ; Hiraoka, Shirai, & Trinh, ; Wasserman, , and the references therein). In recent years, there have appeared a few papers aiming to investigate asymptotic properties of topological signatures, and among such results are of Owada and Thomas (); Yogeshwaran, Subag, and Adler (); and Krebs and Polonik () who focus on limit theorems for Betti numbers for Poisson, binomial, and general stationary point processes. Another notable effort in this research direction has been recently undertaken by Krebs (), who considers asymptotic properties of Betti numbers for generally dependent processes under the assumption of the Marton couplings type of mixing.…”