{
  "id": "1808.01079",
  "version": "v1",
  "published": "2018-08-03T03:45:22.000Z",
  "updated": "2018-08-03T03:45:22.000Z",
  "title": "A fractal dimension for measures via persistent homology",
  "authors": [
    "Henry Adams",
    "Manuchehr Aminian",
    "Elin Farnell",
    "Michael Kirby",
    "Chris Peterson",
    "Joshua Mirth",
    "Rachel Neville",
    "Patrick Shipman",
    "Clayton Shonkwiler"
  ],
  "categories": [
    "math.DS",
    "math.AT",
    "math.PR"
  ],
  "abstract": "We use persistent homology in order to define a family of fractal dimensions, denoted $\\mathrm{dim}_{\\mathrm{PH}}^i(\\mu)$ for each homological dimension $i\\ge 0$, assigned to a probability measure $\\mu$ 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 $\\mu$ is supported on a compact subset of Euclidean space $\\mathbb{R}^m$ for $m\\ge2$, then Steele's work implies that $\\mathrm{dim}_{\\mathrm{PH}}^0(\\mu)=m$ if the absolutely continuous part of $\\mu$ has positive mass, and otherwise $\\mathrm{dim}_{\\mathrm{PH}}^0(\\mu)<m$. 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 $0$-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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-08-03T03:45:22.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "fractal dimension",
      "dimensional persistent homology interval lengths",
      "dimensional homology",
      "euclidean space",
      "probability measure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}