{
  "id": "1707.00087",
  "version": "v1",
  "published": "2017-07-01T02:44:53.000Z",
  "updated": "2017-07-01T02:44:53.000Z",
  "title": "Sharp asymptotic and finite-sample rates of convergence of empirical measures in Wasserstein distance",
  "authors": [
    "Jonathan Weed",
    "Francis Bach"
  ],
  "categories": [
    "math.PR",
    "math.ST",
    "stat.TH"
  ],
  "abstract": "The Wasserstein distance between two probability measures on a metric space is a measure of closeness with applications in statistics, probability, and machine learning. In this work, we consider the fundamental question of how quickly the empirical measure obtained from $n$ independent samples from $\\mu$ approaches $\\mu$ in the Wasserstein distance of any order. We prove sharp asymptotic and finite-sample results for this rate of convergence for general measures on general compact metric spaces. Our finite-sample results show the existence of multi-scale behavior, where measures can exhibit radically different rates of convergence as $n$ grows.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-07-01T02:44:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60B10",
      "62E17"
    ],
    "keywords": [
      "wasserstein distance",
      "sharp asymptotic",
      "empirical measure",
      "finite-sample rates",
      "convergence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}