Papers › A fractal dimension for measures via persistent homology
A fractal dimension for measures via persistent homology
Henry Adams, Manuchehr Aminian, Elin Farnell, Michael Kirby, Chris Peterson, Joshua Mirth, Rachel Neville, Patrick Shipman, Clayton Shonkwiler
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We use persistent homology in order to define a family of fractal dimensions, denoted dim_(PH)ⁱ(μ) for each homological dimension i≥0, assigned to a probability measure μ on a metric space. The case of 0-dimensional homology (i=0) relates to work by Michael J Steele (1988) studying the total length of a minimal spanning tree on a random sampling of points. Indeed, if μ is supported on a compact subset of Euclidean space ℝᵐ for m≥2, then Steele's work implies that dim_(PH)⁰(μ)=m if the absolutely continuous part of μ has positive mass, and otherwise dim_(PH)⁰(μ)<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 μ in an i.i.d. fashion. To some measures μ, 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 0-dimensional homology when μ is the uniform distribution over the unit interval, and conjecture that it exists when μ is the rescaled probability measure for a compact set in Euclidean space with positive Lebesgue measure.
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