{
  "id": "1608.03863",
  "version": "v1",
  "published": "2016-08-12T18:08:06.000Z",
  "updated": "2016-08-12T18:08:06.000Z",
  "title": "Large deviations for random projections of $\\ell_p^n$-balls -- the complete picture",
  "authors": [
    "David Alonso-Gutiérrez",
    "Joscha Prochno",
    "Christoph Thaele"
  ],
  "categories": [
    "math.PR",
    "math.FA"
  ],
  "abstract": "The paper provides a complete description of the large deviation behavior for the Euclidean norm of projections of $\\ell_p^n$-balls to random subspaces of any dimension. More precisely, for each integer $n\\geq 1$, let $k_n\\in\\{1,\\ldots,n-1\\}$, $E^{(n)}$ be a uniform random $k_n$-dimensional subspace of $\\mathbb R^n$ and $X^{(n)}$ be a random point that is uniformly distributed in the $\\ell_p^n$-ball of $\\mathbb R^n$ for some $p\\in[1,\\infty]$. Then the Euclidean norms $\\|P_{E^{(n)}}X^{(n)}\\|_2$ of the orthogonal projections are shown to satisfy a large deviation principle (LDP), as the space dimension $n$ tends to infinity. Its speed and rate function are identified, making thereby visible how they depend on $p$ and the growth of the sequence of subspace dimensions $k_n$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-08-12T18:08:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F10",
      "52A23",
      "60D05",
      "46B09"
    ],
    "keywords": [
      "complete picture",
      "random projections",
      "euclidean norm",
      "large deviation principle",
      "large deviation behavior"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}