{
  "id": "1307.1123",
  "version": "v2",
  "published": "2013-07-03T19:57:54.000Z",
  "updated": "2014-03-01T16:02:29.000Z",
  "title": "The Topology of Probability Distributions on Manifolds",
  "authors": [
    "Omer Bobrowski",
    "Sayan Mukherjee"
  ],
  "categories": [
    "math.PR",
    "math.AT",
    "math.ST",
    "stat.TH"
  ],
  "abstract": "Let $P$ be a set of $n$ random points in $R^d$, generated from a probability measure on a $m$-dimensional manifold $M \\subset R^d$. In this paper we study the homology of $U(P,r)$ -- the union of $d$-dimensional balls of radius $r$ around $P$, as $n \\to \\infty$, and $r \\to 0$. In addition we study the critical points of $d_P$ -- the distance function from the set $P$. These two objects are known to be related via Morse theory. We present limit theorems for the Betti numbers of $U(P,r)$, as well as for number of critical points of index $k$ for $d_P$. Depending on how fast $r$ decays to zero as $n$ grows, these two objects exhibit different types of limiting behavior. In one particular case ($n r^m > C \\log n$), we show that the Betti numbers of $U(P,r)$ perfectly recover the Betti numbers of the original manifold $M$, a result which is of significant interest in topological manifold learning.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-03-01T16:02:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60D05",
      "60F15",
      "60G55",
      "55U10"
    ],
    "keywords": [
      "probability distributions",
      "betti numbers",
      "critical points",
      "dimensional manifold",
      "significant interest"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1307.1123B"
    }
  }
}