Fractal Dimension and the Persistent Homology of Random Geometric Complexes

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.