{
  "id": "2004.05873",
  "version": "v1",
  "published": "2020-04-13T11:35:41.000Z",
  "updated": "2020-04-13T11:35:41.000Z",
  "title": "Analysis of The Ratio of $\\ell_1$ and $\\ell_2$ Norms in Compressed Sensing",
  "authors": [
    "Yiming Xu",
    "Akil Narayan",
    "Hoang Tran",
    "Clayton Webster"
  ],
  "comment": "24 pages, 5 figures",
  "categories": [
    "math.NA",
    "cs.CV",
    "cs.NA",
    "math.OC"
  ],
  "abstract": "We first propose a novel criterion that guarantees that an $s$-sparse signal is the local minimizer of the $\\ell_1/\\ell_2$ objective; our criterion is interpretable and useful in practice. We also give the first uniform recovery condition using a geometric characterization of the null space of the measurement matrix, and show that this condition is easily satisfied for a class of random matrices. We also present analysis on the stability of the procedure when noise pollutes data. Numerical experiments are provided that compare $\\ell_1/\\ell_2$ with some other popular non-convex methods in compressed sensing. Finally, we propose a novel initialization approach to accelerate the numerical optimization procedure. We call this initialization approach \\emph{support selection}, and we demonstrate that it empirically improves the performance of existing $\\ell_1/\\ell_2$ algorithms.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-04-13T11:35:41.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "compressed sensing",
      "first uniform recovery condition",
      "novel initialization approach",
      "popular non-convex methods",
      "noise pollutes data"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}