{
  "id": "2307.01181",
  "version": "v1",
  "published": "2023-07-03T17:46:23.000Z",
  "updated": "2023-07-03T17:46:23.000Z",
  "title": "Fitting an ellipsoid to a quadratic number of random points",
  "authors": [
    "Afonso S. Bandeira",
    "Antoine Maillard",
    "Shahar Mendelson",
    "Elliot Paquette"
  ],
  "comment": "17 pages",
  "categories": [
    "math.PR",
    "cs.DS",
    "cs.LG",
    "math.ST",
    "stat.ML",
    "stat.TH"
  ],
  "abstract": "We consider the problem $(\\mathrm{P})$ of fitting $n$ standard Gaussian random vectors in $\\mathbb{R}^d$ to the boundary of a centered ellipsoid, as $n, d \\to \\infty$. This problem is conjectured to have a sharp feasibility transition: for any $\\varepsilon > 0$, if $n \\leq (1 - \\varepsilon) d^2 / 4$ then $(\\mathrm{P})$ has a solution with high probability, while $(\\mathrm{P})$ has no solutions with high probability if $n \\geq (1 + \\varepsilon) d^2 /4$. So far, only a trivial bound $n \\geq d^2 / 2$ is known on the negative side, while the best results on the positive side assume $n \\leq d^2 / \\mathrm{polylog}(d)$. In this work, we improve over previous approaches using a key result of Bartl & Mendelson on the concentration of Gram matrices of random vectors under mild assumptions on their tail behavior. This allows us to give a simple proof that $(\\mathrm{P})$ is feasible with high probability when $n \\leq d^2 / C$, for a (possibly large) constant $C > 0$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-07-03T17:46:23.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "random points",
      "quadratic number",
      "high probability",
      "standard gaussian random vectors",
      "sharp feasibility transition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}