2018/08/31 by Henry Adams, Manuchehr Aminian, Elin Farnell +6
Computer Science · Mathematics · #Absolute continuity #Combinatorics #Discrete mathematics #Euclidean space #Fractal #Homology (biology) #Homotopy and Cohomology in Algebraic Topology #Lebesgue integration #Lebesgue measure #Mathematical analysis #Mathematics #Probability measure #Topological and Geometric Data Analysis #math.AT #math.DS #math.PR
paper · pdf · doi:10.1007/978-3-030-43408-3_1
published as In: Baas N., Carlsson G., Quick G., Szymik M., Thaule M. (eds), Topological Data Analysis. Abel Symposia, vol 15. Springer (2020)
arxiv created 2019/01/30 · openalex publication_date 2020/01/01 · arxiv updated 2020/07/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We use persistent homology in order to define a family of fractal dimensions, denoted dimPHi(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 dimPH0(μ)=m if the absolutely continuous part of μ has positive mass, and otherwise dimPH0(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.