[Paper Review] Limit theorems for process-level Betti numbers for sparse, critical, and Poisson regimes

Takashi Owada, Andrew M. Thomas|arXiv (Cornell University)|Sep 15, 2018

Topological and Geometric Data Analysis18 references4 citations

TL;DR

This paper establishes limit theorems for process-level Betti numbers of Čech complexes built from Poisson-distributed points in $[ d$-dimensional Euclidean space, analyzing sparse, critical, and Poisson regimes. It proves a central limit theorem for sparse and critical regimes (with limiting processes represented as time-changed Brownian motion and Gaussian processes, respectively) and a Poisson limit theorem for the Poisson regime, providing a complete asymptotic characterization of topological complexity in random geometric complexes.

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 Čech complex causes simplices of dimension greater than $k+1$ to vanish in probability, the so-called sparse and Poisson regimes, as well when the connectivity radius is on the order of $n^{-1/d}$, the critical regime. We establish limit theorems in all of the aforementioned regimes, a central limit theorem for the sparse and critical regimes, and a Poisson limit theorem for the Poisson regime. When the connectivity radius of the Čech complex is $o(n^{-1/d})$, i.e., the sparse and Poisson regimes, we can decompose the limiting processes into a time-changed Brownian motion and a time-changed homogeneous Poisson process respectively. In the critical regime, the limiting process is a centered Gaussian process but has much more complicated representation, because the Čech complex becomes highly connected with many topological holes of any dimension.

Motivation & Objective

• To analyze the asymptotic behavior of Betti numbers as stochastic processes in random geometric topology.
• To characterize the limiting distributions of process-level Betti numbers under three distinct scaling regimes: sparse, critical, and Poisson.
• To establish central limit theorems for the sparse and critical regimes and a Poisson limit theorem for the Poisson regime.
• To derive explicit representations of the limiting processes, including time-changed Brownian motion and Gaussian processes.
• To understand the topological complexity of Čech complexes formed from Poisson processes with intensity $nf$ in $[ d$-dimensional space.

Proposed method

• Model the Čech complex using a Poisson point process with intensity $nf$, where $f$ is a probability density on $[ d$.
• Analyze Betti numbers $\beta_k$ as stochastic processes indexed by the connectivity radius $t$, tracking topological features across scales.
• Use moment methods and factorial moment measures to control the behavior of higher-order simplices and their contributions to Betti numbers.
• Apply the continuous mapping theorem and Laplace functional of Poisson random measures to characterize the weak convergence of the limiting processes.
• Employ change-of-variables and bounds on integrals involving indicator functions of simplex intersections to prove vanishing variance terms.
• Establish convergence of expectations and second moments via asymptotic expansions and bounds on the probability of complex connectivity.

Experimental results

Research questions

• RQ1 How do Betti numbers of Čech complexes behave as stochastic processes when the underlying point process is a Poisson process with intensity $nf$?
• RQ2 What is the limiting distribution of the $k$-th Betti number process in the sparse regime, where the connectivity radius is $o(n^{-1/d})$?
• RQ3 What is the limiting behavior of the Betti number process in the critical regime, where the connectivity radius is on the order of $n^{-1/d}$?
• RQ4 How does the limiting process differ in the Poisson regime compared to the sparse and critical regimes?
• RQ5 Can the limiting process in the critical regime be represented explicitly, and what is its dependence on the geometry of the underlying space?

Key findings

• In the sparse regime, the limiting process for Betti numbers is a time-changed Brownian motion, reflecting diffusive fluctuations in topological features.
• In the critical regime, the limiting process is a centered Gaussian process with a complex representation due to high connectivity and persistent topological holes of all dimensions.
• In the Poisson regime, the limiting process converges weakly to a time-changed homogeneous Poisson process, indicating rare, independent topological events.
• The expectation of the $k$-th Betti number process converges to a deterministic limit given by $\mu_{k,\mathbb{R}^d}(t,t)$, which captures the expected number of $k$-cycles.
• The remainder term $R_{k,n}(t)$, representing contributions from high-dimensional simplices, vanishes in expectation as $n \to \infty$, ensuring convergence of the main term.
• The convergence of the finite-dimensional distributions is established via moment convergence and the continuous mapping theorem applied to Poisson random measures.