{
  "id": "1802.08703",
  "version": "v1",
  "published": "2018-02-23T19:21:33.000Z",
  "updated": "2018-02-23T19:21:33.000Z",
  "title": "Large data limit for a phase transition model with the p-Laplacian on point clouds",
  "authors": [
    "Riccardo Cristoferi",
    "Matthew Thorpe"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "The consistency of a nonlocal anisotropic Ginzburg-Landau type functional for data classification and clustering is studied. The Ginzburg-Landau objective functional combines a double well potential, that favours indicator valued function, and the $p$-Laplacian, that enforces regularity. Under appropriate scaling between the two terms minimisers exhibit a phase transition on the order of $\\epsilon=\\epsilon_n$ where $n$ is the number of data points. We study the large data asymptotics, i.e. as $n\\to \\infty$, in the regime where $\\epsilon_n\\to 0$. The mathematical tool used to address this question is $\\Gamma$-convergence. In particular, it is proved that the discrete model converges to a weighted anisotropic perimeter.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-02-23T19:21:33.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "phase transition model",
      "large data limit",
      "point clouds",
      "nonlocal anisotropic ginzburg-landau type functional",
      "p-laplacian"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}