{
  "id": "2007.15263",
  "version": "v1",
  "published": "2020-07-30T07:03:37.000Z",
  "updated": "2020-07-30T07:03:37.000Z",
  "title": "A projected gradient method for $α\\ell_{1}-β\\ell_{2}$ sparsity regularization",
  "authors": [
    "Liang Ding",
    "Weimin Han"
  ],
  "comment": "30 pages; 8 figures",
  "categories": [
    "math.NA",
    "cs.NA"
  ],
  "abstract": "The non-convex $\\alpha\\|\\cdot\\|_{\\ell_1}-\\beta\\| \\cdot\\|_{\\ell_2}$ $(\\alpha\\ge\\beta\\geq0)$ regularization has attracted attention in the field of sparse recovery. One way to obtain a minimizer of this regularization is the ST-($\\alpha\\ell_1-\\beta\\ell_2$) algorithm which is similar to the classical iterative soft thresholding algorithm (ISTA). It is known that ISTA converges quite slowly, and a faster alternative to ISTA is the projected gradient (PG) method. However, the conventional PG method is limited to the classical $\\ell_1$ sparsity regularization. In this paper, we present two accelerated alternatives to the ST-($\\alpha\\ell_1-\\beta\\ell_2$) algorithm by extending the PG method to the non-convex $\\alpha\\ell_1-\\beta\\ell_2$ sparsity regularization. Moreover, we discuss a strategy to determine the radius $R$ of the $\\ell_1$-ball constraint by Morozov's discrepancy principle. Numerical results are reported to illustrate the efficiency of the proposed approach.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-07-30T07:03:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "65K10",
      "G.1.6"
    ],
    "keywords": [
      "sparsity regularization",
      "projected gradient method",
      "iterative soft thresholding algorithm",
      "morozovs discrepancy principle",
      "conventional pg method"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}