Fractal Dimension and the Persistent Homology of Random Geometric Complexes

TL;DR

This paper demonstrates that the fractal dimension of a metric space with an Ahlfors regular measure can be inferred from the persistent homology of random samples, extending classical results to fractal settings.

Contribution

It generalizes Steele's 1988 result to fractal measures, linking persistent homology and fractal dimension through probabilistic analysis.

Findings

The $ ext{E}_ ext{alpha}^0$ sum scales with sample size as predicted by the fractal dimension.

The logarithm of the $ ext{E}_ ext{alpha}^0$ sum converges to a value related to the fractal dimension.

Analogous results are established for higher-dimensional persistent homology sums.

Abstract

We prove that the fractal dimension of a metric space equipped with an Ahlfors regular measure can be recovered from the persistent homology of random samples. Our main result is that if $x_1,\ldots, x_n$ are i.i.d. samples from a $d$-Ahlfors regular measure on a metric space, and $E^0_\alpha\left(x_1,\ldots,x_n\right)$ denotes the $\alpha$-weight of the minimum spanning tree on $x_1,\ldots,x_n:$ \[E_\alpha^0\left(x_1,\ldots,x_n\right)=\sum_{e\in T\left(x_1,\ldots,x_n\right)} |e|^\alpha\,,\] then there exist constants $0<C_1\leq C_2$ so that \[C_1\leq n^{-\frac{d-\alpha}{d}} E^0_\alpha\left(x_1,\ldots,x_n\right)\leq C_2\,\] with high probability as $n\rightarrow \infty.$ In particular, \[\log\big(E^0_\alpha(x_1,\ldots,x_n)\big)/\log(n)\longrightarrow (d-\alpha)/d\,.\] This is a generalization of a result of Steele (1988) from the non-singular case to the fractal setting. Our result is best possible, in the sense that there exist Ahlfors regular measures for which the limit $\lim_{n\rightarrow\infty} n^{-\frac{d-\alpha}{d}} E^0_\alpha\left(x_1,\ldots,x_n\right)$ does not exist with high probability. We also prove analogous results for weighted sums defined in terms of higher dimensional persistent homology.