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), ..., x(n) are i.i.d. samples from a d-Ahlfors regular measure on a metric space, and E-alpha(0)(x(1), ..., x(n)) denotes the alpha-weight of the minimum spanning tree on x(1), ..., x(n): E-alpha(0)(x(1), ..., x(n)) = Sigma(e is an element of T(x1,...,xn)) vertical bar e vertical bar(alpha), then there exist constants 0 < C-1 <= C-2 so that C-1 <= n(-d-alpha/d) E-alpha(0)(x(1), ..., x(n)) <= C-2 with high probability as n -> infinity. In particular, d can be recovered from the limit log(E-alpha(0)(x(1), ..., x(n))) / log(n) -> (d - alpha)/d This is a generalization of a result of Steele [62] from the non-singular case to the fractal setting. We also construct an example of an Ahlfors regular measure for which the limit lim(n ->infinity) n(-d-alpha/d) E-alpha(0)(x(1), ..., x(n)) does not exist with high probability, and prove analogous results for weighted sums defined in terms of higher dimensional persistent homology. (C) 2020 Elsevier Inc. All rights reserved.