[Paper Review] Weighted Persistent Homology Sums of Random \v{C}ech Complexes

Benjamin Schweinhart|arXiv (Cornell University)|Jul 18, 2018

Topological and Geometric Data Analysis17 references3 citations

TL;DR

This paper establishes the asymptotic behavior of weighted persistent homology sums in random Čech complexes built from i.i.d. samples on metric spaces. It proves that for points sampled uniformly from an m-dimensional sphere or ball, the α-weighted sum of i-dimensional persistent homology intervals scales like n^{(m−α)/m} in expectation, with tight bounds in probability, generalizing classical results on minimal spanning trees to higher-dimensional topological features.

ABSTRACT

We study the asymptotic behavior of random variables of the form \\begin{equation*} E_{\\alpha}^i\\left(x_1,\\ldots,x_n\ ight)=\\sum_{\\left(b,d\ ight)\\in \\mathit{PH}_i\\left(x_1,\\ldots,x_n\ ight)} \\left(d-b\ ight)^{\\alpha} \\end{equation*} where $\\left\\{x_j\ ight\\}_{j\\in\\mathbb{N}}$ are i.i.d. samples from a probability measure on a triangulable metric space, and $\ extit{PH}_i\\left(x_1,\\ldots,x_n\ ight)$ denotes the $i$-dimensional reduced persistent homology of the \\v{C}ech complex of $\\left\\{x_1,\\ldots,x_n\ ight\\}.$ These quantities are a higher-dimensional generalization of the $\\alpha$-weighted sum of a minimal spanning tree; we seek to prove analogues of the theorems of Steele (1988) and Aldous and Steele (1992) in this context. As a special case of our main theorem, we show that if $\\left\\{x_j\ ight\\}_{j\\in\\mathbb{N}}$ are distributed independently and uniformly on the $m$-dimensional Euclidean sphere, $\\alpha<m,$ and $0\\leq i <n,$ then there are real numbers $\\gamma$ and $\\Gamma$ so that \\begin{equation*} \\gamma \\leq \\lim_{n\ ightarrow\\infty} n^{-\\frac{m-\\alpha}{m}} E_i^{\\alpha}\\left(x_1,\\ldots,x_n\ ight) \\leq \\Gamma \\end{equation*} in probability. More generally, we prove results about the asymptotics of the expectation of $E_\\alpha^i$ for points sampled from a locally bounded probability measure on a space that is the bi-Lipschitz image of an $m-$dimensional Euclidean simplicial complex, as well as measures supported on sets of fractional dimension that respect box counting.

Motivation & Objective

• To generalize classical probabilistic results on minimal spanning trees to higher-dimensional topological features via persistent homology.
• To analyze the asymptotic behavior of α-weighted sums of persistent homology intervals in random Čech complexes.
• To establish almost-sure and in-probability convergence bounds for these sums under sampling from compact, locally bounded measures on m-dimensional spaces.
• To extend known extremal bounds on persistent homology to probabilistic settings with explicit scaling laws.

Proposed method

• Uses weighted persistent homology sums defined as $ E_{eta}^{i} = \sum_{(b,d) \in \mathrm{PH}_i} (d-b)^\alpha $, generalizing minimal spanning tree weights.
• Applies bi-Lipschitz invariance of persistent homology to relate sampling on curved manifolds to Euclidean models.
• Employs probabilistic concentration arguments and Poisson approximation to control the number of intervals in given lifetime bands.
• Derives lower and upper bounds via comparison with Poisson processes and volume estimates in small balls.
• Uses interleaving lemmas for bi-Lipschitz maps to transfer results from Euclidean space to general m-dimensional metric spaces.
• Applies results from Cohen-Steiner et al. on uniform boundedness of $ E_{\alpha}^{i} $ when $ \alpha > m $ to establish upper bounds.

Experimental results

Research questions

• RQ1 What is the asymptotic scaling of the α-weighted sum of persistent homology intervals in random Čech complexes on an m-dimensional sphere?
• RQ2 Does the limit $ \lim_{n \to \infty} n^{-(m-\alpha)/m} E_{\alpha}^{i}(x_1,\dots,x_n) $ exist and remain bounded in probability for $ \alpha < m $?
• RQ3 How does the behavior of $ E_{\alpha}^{i} $ on a sphere or ball compare to classical results on minimal spanning trees in $ \mathbb{R}^m $?
• RQ4 Can the scaling law for $ E_{\alpha}^{i} $ be extended beyond $ i=0 $ to higher-dimensional homology groups?
• RQ5 What is the behavior of $ E_{m}^{i} $, the total lifetime persistence, as $ n \to \infty $?

Key findings

• For i.i.d. uniform samples on the m-dimensional sphere, $ \gamma \leq \lim_{n \to \infty} n^{-(m-\alpha)/m} E_{\alpha}^{i} \leq \Gamma $ in probability for $ \alpha < m $, with constants $ \gamma, \Gamma $ depending on $ \alpha $ and the measure.
• On the m-dimensional Euclidean ball, the same scaling holds in expectation: $ \gamma \leq \lim_{n \to \infty} n^{-(m-\alpha)/m} \mathbb{E}[E_{\alpha}^{i}] \leq \Gamma $, with the lower bound also holding in probability.
• For $ \alpha = m $, the total lifetime persistence satisfies $ \lim_{n \to \infty} \frac{1}{\log n} \mathbb{E}[E_{m}^{i}] \leq D $ in probability for some finite $ D $.
• The lower bound is established via a Poisson approximation argument, showing that $ \lim_{n \to \infty} n^{-(m-\alpha)/m} E_{\alpha}^{i} \geq \gamma' $ in probability for some $ \gamma' > 0 $.
• The results extend to locally bounded probability measures on bi-Lipschitz images of m-dimensional simplicial complexes, using interleaving lemmas for persistent homology under bi-Lipschitz maps.
• The paper generalizes Steele’s theorem on minimal spanning trees and Aldous–Steele’s $ L^2 $ convergence to higher-dimensional persistent homology, establishing a new class of scaling laws in topological data analysis.