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A fractal dimension for measures via persistent homology. (English) Zbl 1448.62211
Baas, Nils (ed.) et al., Topological data analysis. Proceedings of the Abel symposium 2018, Geiranger, Norway, June 4–8, 2018. Cham: Springer. Abel Symp. 15, 1-31 (2020).
Summary: We use persistent homology in order to define a family of fractal dimensions, denoted $$\dim_{\text{PH}}^i(\mu)$$ for each homological dimension $$i\geq 0$$, assigned to a probability measure $$\mu$$ on a metric space. The case of zero-dimensional homology $$(i = 0)$$ relates to work by J. M. Steele [Ann. Probab. 16, No. 4, 1767–1787 (1988; Zbl 0655.60023)] studying the total length of a minimal spanning tree on a random sampling of points. Indeed, if $$\mu$$ is supported on a compact subset of Euclidean space $$\mathbb{R}^m$$ for $$m \geq 2$$, then Steele’s work implies that $$\dim_{\text{PH}}^0(\mu )=m$$ if the absolutely continuous part of $$\mu$$ has positive mass, and otherwise $$\dim_{\text{PH}}^0(\mu )<m$$. Experiments suggest that similar results may be true for higher-dimensional homology $$0<i<m$$, though this is an open question. Our fractal dimension is defined by considering a limit, as the number of points $$n$$ goes to infinity, of the total sum of the $$i$$-dimensional persistent homology interval lengths for $$n$$ random points selected from $$\mu$$ in an i.i.d. fashion. To some measures $$\mu$$, we are able to assign a finer invariant, a curve measuring the limiting distribution of persistent homology interval lengths as the number of points goes to infinity. We prove this limiting curve exists in the case of zero-dimensional homology when $$\mu$$ is the uniform distribution over the unit interval, and conjecture that it exists when $$\mu$$ is the rescaled probability measure for a compact set in Euclidean space with positive Lebesgue measure.
For the entire collection see [Zbl 1448.62008].

##### MSC:
 62R40 Topological data analysis 62R20 Statistics on metric spaces 55N31 Persistent homology and applications, topological data analysis 60B05 Probability measures on topological spaces 37F35 Conformal densities and Hausdorff dimension for holomorphic dynamical systems 60F15 Strong limit theorems
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GitHub; javaPlex; LearningAlgebraicVarieties; plfit; Ripser
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