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%DIFDELCMD < equation* 1{c} r^d \leq \mu \big(B_r\left(x\right)\big) \leq c r^d equation* %%% for some c>0, all x in the support of \mu, and all sufficiently small r . Our main result is that if x_1,\ldots  x_n are sampled from a d-Ahlfors regular measure on \mathbb{R and E_\alpha\left(x_1,\ldots,x_n\right) denotes the \alpha-weight of the minimal spanning tree on x_1,\ldots,x_n: %DIFDELCMD < \[E_\alpha\left(x_1,\ldots,x_n\right)=e\in T\left(x_1,\ldots,x_n\right) |e|^\alpha\] %%% then%DIFDELCMD < \[E_\alpha\left(x_1,\ldots,x_n\right) \approx n^{d-\alpha{d}}\] %%% \[E_\alpha\left(x_1,\ldots,x_n\right)=\sum_{e\in T\left(x_1,\ldots,x_n\right)} |e|^\alpha\,,\] \[E_\alpha\left(x_1,\ldots,x_n\right)\approx n^{\frac{d-\alpha}{d}}\] with high probability as n\rightarrow \infty. \[\log\big(E_\alpha(x_1,\ldots,x_n)\big)/\log(n)\longrightarrow (d-\alpha)/d\,.\]  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) .
We %DIFDELCMD < equation* 1{c} r^d \leq \mu \big(B_r\left(x\right)\big) \leq c r^d equation* %%% 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_\alpha\left(x_1,\ldots,x_n\right) denotes the \alpha-weight of the minimum spanning tree on x_1,\ldots,x_n: %DIFDELCMD < \[E_\alpha\left(x_1,\ldots,x_n\right)=e\in T\left(x_1,\ldots,x_n\right) |e|^\alpha\] %%% %DIFDELCMD < \[E_\alpha\left(x_1,\ldots,x_n\right) \approx n^{d-\alpha{d}}\] %%% \[E_\alpha\left(x_1,\ldots,x_n\right)=\sum_{e\in T\left(x_1,\ldots,x_n\right)} |e|^\alpha\,,\] then\[E_\alpha\left(x_1,\ldots,x_n\right)\approx n^{\frac{d-\alpha}{d}}\] with high probability as n\rightarrow \infty. In particular,\[\log\big(E_\alpha(x_1,\ldots,x_n)\big)/\log(n)\longrightarrow (d-\alpha)/d\,.\] This is a generalization of a result of Steele (1988) from the absolutely continuous case to the fractal setting. We also prove analogous results for weighted sums defined in terms of higher dimensional persistent homology .
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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
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for some c>0, all x in the support of \mu, and all sufficiently small r
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
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,
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sampled
i.i.d.~samples
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\mathbb{R
a metric space,
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minimal
minimum
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then
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then
A
In particular,
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This is a generalization of a result of Steele (1988) from the absolutely continuous case to the fractal setting.
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theorems about the asymptotic behavior of
analogous results for
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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)
higher dimensional persistent homology