{
  "id": "1604.03364",
  "version": "v1",
  "published": "2016-04-12T12:19:57.000Z",
  "updated": "2016-04-12T12:19:57.000Z",
  "title": "Elastic-net regularization versus $\\ell^1$-regularization for linear inverse problems with quasi-sparse solutions",
  "authors": [
    "De-Han Chen",
    "Bernd Hofmann",
    "Jun Zou"
  ],
  "comment": "16 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "We consider the ill-posed operator equation $Ax=y$ with an injective and bounded linear operator $A$ mapping between $\\ell^2$ and a Hilbert space $Y$, and a unique solution $x^\\dag=\\{x^\\dag_k\\}_{k=1}^\\infty$. For the cases that sparsity $x^\\dag \\in \\ell^0$ is expected but often slightly violated in practice, we investigate, in comparison with the $\\ell^1$-regularization, the elastic-net regularization, where the penalty is a weighted superposition of the $\\ell^1$-norm and the $\\ell^2$-norm square, under the assumption $x^\\dag \\in \\ell^1$. There occur two positive parameters in this approach, the weight parameter $\\eta$ and the regularization parameter as the multiplier of the whole penalty in the Tikhonov functional, whereas only one regularization parameter arises in $\\ell^1$-regularization. Based on the variational inequality approach for the description of the solution smoothness with respect to the forward operator $A$ and exploiting the method of approximate source conditions we present some results to estimate the rate of convergence for the elastic-net regularization. The occurring rate function contains the rate of the decay $x^\\dag \\to 0$ for $k \\to \\infty$ and the classical smoothness properties of $x^\\dag$ as an element in $\\ell^2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-04-12T12:19:57.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "linear inverse problems",
      "elastic-net regularization",
      "quasi-sparse solutions",
      "regularization parameter arises",
      "variational inequality approach"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160403364C"
    }
  }
}