[Paper Review] Betti Numbers of Gaussian Excursions in the Sparse Regime

Gugan Thoppe, Sunder Ram Krishnan|arXiv (Cornell University)|Jul 29, 2018

Topological and Geometric Data Analysis25 references3 citations

TL;DR

This paper presents the first comprehensive analysis of Betti numbers—topological invariants quantifying connected components and holes—in Gaussian field excursion sets within the sparse regime. By modeling excursions via Čech complexes and combining extreme value theory with combinatorial topology, the authors derive limit theorems for Betti numbers, including phase transitions, Poisson approximations, and central limit theorems near critical thresholds, under minimal assumptions on covariance structure.

ABSTRACT

Random field excursions is an increasingly vital topic within data analysis in medicine, cosmology, materials science, etc. This work is the first detailed study of their Betti numbers in the so-called `sparse' regime. Specifically, we consider a piecewise constant Gaussian field whose covariance function is positive and satisfies some local, boundedness, and decay rate conditions. We model its excursion set via a Cech complex. For Betti numbers of this complex, we then prove various limit theorems as the window size and the excursion level together grow to infinity. Our results include asymptotic mean and variance estimates, a vanishing to non-vanishing phase transition with a precise estimate of the transition threshold, and a weak law in the non-vanishing regime. We further obtain a Poisson approximation and a central limit theorem close to the transition threshold. Our proofs combine extreme value theory and combinatorial topology tools.

Motivation & Objective

• To provide the first detailed study of Betti numbers in the sparse regime for Gaussian field excursions.
• To establish asymptotic mean and variance estimates for Betti numbers of excursion sets modeled via Čech complexes.
• To identify the precise threshold for a phase transition from vanishing to non-vanishing Betti numbers in the sparse regime.
• To derive weak laws, Poisson approximations, and central limit theorems for Betti numbers near the critical threshold.
• To extend topological data analysis tools to sparse Gaussian fields using minimal assumptions on covariance decay.

Proposed method

• Model the excursion set of a piecewise constant, zero-mean, stationary Gaussian field on a discrete lattice using a Čech complex.
• Use combinatorial topology tools to approximate Betti numbers of the complex via local statistics on lattice sites.
• Apply the Stein-Chen method for Poisson approximation to control rare events in extreme field values.
• Leverage Slepian’s lemma and Savage’s multivariate Gaussian tail estimates to bound tail probabilities of field maxima.
• Establish convergence rates for Betti number distributions by combining extreme value theory with spectral bounds on covariance matrices.
• Transfer results from approximating statistics to the true Betti numbers using topological stability arguments from persistent homology.

Experimental results

Research questions

• RQ1 What is the asymptotic behavior of Betti numbers βk for high-level excursions of Gaussian fields in the sparse regime?
• RQ2 At what threshold does the expected number of k-dimensional holes transition from vanishing to non-vanishing as the window size and level grow?
• RQ3 How do Betti numbers behave near the critical threshold—can they be approximated by a Poisson or normal distribution?
• RQ4 What are the precise rates of convergence for the mean and variance of Betti numbers in the non-vanishing regime?
• RQ5 To what extent can the results be obtained under weak assumptions on the covariance function, such as integrability rather than exponential decay?

Key findings

• The paper establishes a precise threshold for the phase transition from vanishing to non-vanishing Betti numbers in the sparse regime, derived from the covariance decay rate and field level.
• Asymptotic mean and variance estimates for Betti numbers βk are derived, showing convergence to a non-degenerate limit under appropriate scaling.
• A weak law of large numbers is proven for Betti numbers in the non-vanishing regime, confirming concentration around the limiting mean.
• Near the critical threshold, a Poisson approximation is established for the number of isolated components (β0), indicating rare-event behavior.
• A central limit theorem is derived for Betti numbers in the vicinity of the phase transition, validating Gaussian fluctuations in the critical window.
• The results hold under minimal assumptions: the covariance function is positive, locally bounded, and integrable, with no requirement for exponential decay.