Fractal Dimension and the Persistent Homology of Random Geometric Complexes
Benjamin Schweinhart · arXiv (Cornell University) · 2018
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.