Weighted Persistent Homology Sums of Random \v{C}ech Complexes

TL;DR

This paper investigates the asymptotic behavior of weighted sums of persistent homology in random Čech complexes, extending classical results to higher dimensions and more complex spaces.

Contribution

It generalizes the asymptotic analysis of minimal spanning trees to higher-dimensional persistent homology sums in random geometric complexes.

Findings

Established bounds for the asymptotic behavior of weighted persistent homology sums.

Proved convergence results for points on Euclidean spheres and more general spaces.

Extended classical theorems to higher-dimensional topological data analysis contexts.

Abstract

We study the asymptotic behavior of random variables of the form \begin{equation*} E_{\alpha}^i\left(x_1,\ldots,x_n\right)=\sum_{\left(b,d\right)\in \mathit{PH}_i\left(x_1,\ldots,x_n\right)} \left(d-b\right)^{\alpha} \end{equation*} where $\left\{x_j\right\}_{j\in\mathbb{N}}$ are i.i.d. samples from a probability measure on a triangulable metric space, and $\textit{PH}_i\left(x_1,\ldots,x_n\right)$ denotes the $i$-dimensional reduced persistent homology of the \v{C}ech complex of $\left\{x_1,\ldots,x_n\right\}.$ 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 $\left\{x_j\right\}_{j\in\mathbb{N}}$ 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\rightarrow\infty} n^{-\frac{m-\alpha}{m}} E_i^{\alpha}\left(x_1,\ldots,x_n\right) \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.