The Persistent Homology of Random Geometric Complexes on Fractals
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We study the asymptotic behavior of the persistent homology of i.i.d. samples from a d-Ahlfors regular measure --- one that satisfies uniform bounds of the form 1/c r^d ≤μ(B_r(x)) ≤ c r^d for some c>0, all x in the support of μ, and all sufficiently small r. Our main result is that if x_1,... x_n are sampled from a d-Ahlfors regular measure on R^m and E_α(x_1,...,x_n) denotes the α-weight of the minimal spanning tree on x_1,...,x_n: E_α(x_1,...,x_n)=∑_e∈ T(x_1,...,x_n) |e|^α then E_α(x_1,...,x_n) ≈ n^d-α/d with high probability as n→∞. We also prove theorems about the asymptotic behavior of weighted sums defined in terms of higher-dimensional persistent homology. As an application, we exhibit hypotheses under which the fractal dimension of a measure can be computed from the persistent homology of i.i.d. samples from that space, in a manner similar to that proposed in the experimental work of Adams et al. (2018).
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