{
  "id": "2001.00220",
  "version": "v1",
  "published": "2020-01-01T15:13:41.000Z",
  "updated": "2020-01-01T15:13:41.000Z",
  "title": "Statistical Invariance of Betti Numbers in the Thermodynamic Regime",
  "authors": [
    "Siddharth Vishwanath",
    "Kenji Fukumizu",
    "Satoshi Kuriki",
    "Bharath Sriperumbudur"
  ],
  "categories": [
    "math.PR",
    "math.AT",
    "math.ST",
    "stat.TH"
  ],
  "abstract": "Topological Data Analysis (TDA) has become a promising tool to uncover low-dimensional features from high-dimensional data. Notwithstanding the advantages afforded by TDA, its adoption in statistical methodology has been limited by several reasons. In this paper we study the framework of topological inference through the lens of classical parametric inference in statistics. Suppose $\\mathcal{P} = \\{\\mathbb{P}_\\theta : \\theta \\in \\Theta\\}$ is a parametric family of distributions indexed by a set $\\Theta$, and $\\mathbb{X}_n = \\{\\mathbf{X}_1, \\mathbf{X}_2, \\dots, \\mathbf{X}_n\\}$ is observed iid at random from a distribution $\\mathbb{P}_\\theta$. The asymptotic behaviour of the Betti numbers associated with the \\v{C}ech complex of $\\mathbb{X}_n$ contain both the topological and parametric information about the distribution of points. We investigate necessary and sufficient conditions under which topological inference is possible in this parametric setup.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-01-01T15:13:41.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "betti numbers",
      "thermodynamic regime",
      "statistical invariance",
      "uncover low-dimensional features",
      "topological inference"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}