{
  "id": "math/0307152",
  "version": "v2",
  "published": "2003-07-10T21:36:11.000Z",
  "updated": "2003-11-02T19:56:43.000Z",
  "title": "An iterative thresholding algorithm for linear inverse problems with a sparsity constraint",
  "authors": [
    "Ingrid Daubechies",
    "Michel Defrise",
    "Christine De Mol"
  ],
  "comment": "30 pages, 3 figures; this is version 2 - changes with respect to v1: small correction in proof (but not statement of) lemma 3.15; description of Besov spaces in intro and app A clarified (and corrected); smaller pointsize (making 30 instead of 38 pages)",
  "categories": [
    "math.FA",
    "math.NA"
  ],
  "abstract": "We consider linear inverse problems where the solution is assumed to have a sparse expansion on an arbitrary pre-assigned orthonormal basis. We prove that replacing the usual quadratic regularizing penalties by weighted l^p-penalties on the coefficients of such expansions, with 1 < or = p < or =2, still regularizes the problem. If p < 2, regularized solutions of such l^p-penalized problems will have sparser expansions, with respect to the basis under consideration. To compute the corresponding regularized solutions we propose an iterative algorithm that amounts to a Landweber iteration with thresholding (or nonlinear shrinkage) applied at each iteration step. We prove that this algorithm converges in norm. We also review some potential applications of this method.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2003-11-02T19:56:43.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "linear inverse problems",
      "iterative thresholding algorithm",
      "sparsity constraint",
      "arbitrary pre-assigned orthonormal basis",
      "regularized solutions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2003math......7152D"
    }
  }
}