{
  "id": "1406.6712",
  "version": "v2",
  "published": "2014-06-25T20:36:44.000Z",
  "updated": "2015-06-19T12:57:30.000Z",
  "title": "Stability of low-rank matrix recovery and its connections to Banach space geometry",
  "authors": [
    "Javier Alejandro Chávez-Domínguez",
    "Denka Kutzarova"
  ],
  "comment": "16 pages",
  "journal": "J. Math. Anal. Appl. 427 (2015), no. 1, 320--335",
  "doi": "10.1016/j.jmaa.2015.02.041",
  "categories": [
    "math.FA",
    "cs.IT",
    "math.IT"
  ],
  "abstract": "There are well-known relationships between compressed sensing and the geometry of the finite-dimensional $\\ell_p$ spaces. A result of Kashin and Temlyakov can be described as a characterization of the stability of the recovery of sparse vectors via $\\ell_1$-minimization in terms of the Gelfand widths of certain identity mappings between finite-dimensional $\\ell_1$ and $\\ell_2$ spaces, whereas a more recent result of Foucart, Pajor, Rauhut and Ullrich proves an analogous relationship even for $\\ell_p$ spaces with $p < 1$. In this paper we prove what we call matrix or noncommutative versions of these results: we characterize the stability of low-rank matrix recovery via Schatten $p$-(quasi-)norm minimization in terms of the Gelfand widths of certain identity mappings between finite-dimensional Schatten $p$-spaces.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-06-25T20:36:44.000Z",
      "comment": "19 pages, 1 figure",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-06-19T12:57:30.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "low-rank matrix recovery",
      "banach space geometry",
      "connections",
      "identity mappings",
      "gelfand widths"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Elsevier"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1406.6712A"
    }
  }
}