{
  "id": "1310.7637",
  "version": "v2",
  "published": "2013-10-28T22:08:22.000Z",
  "updated": "2014-02-25T14:51:44.000Z",
  "title": "Regularization of $\\ell_1$ minimization for dealing with outliers and noise in Statistics and Signal Recovery",
  "authors": [
    "Salvador Flores",
    "Luis M. Briceno-Arias"
  ],
  "categories": [
    "math.OC",
    "stat.ML"
  ],
  "abstract": "We study the robustness properties of $\\ell_1$ norm minimization for the classical linear regression problem with a given design matrix and contamination restricted to the dependent variable. We perform a fine error analysis of the $\\ell_1$ estimator for measurements errors consisting of outliers coupled with noise. We introduce a new estimation technique resulting from a regularization of $\\ell_1$ minimization by inf-convolution with the $\\ell_2$ norm. Concerning robustness to large outliers, the proposed estimator keeps the breakdown point of the $\\ell_1$ estimator, and reduces to least squares when there are not outliers. We present a globally convergent forward-backward algorithm for computing our estimator and some numerical experiments confirming its theoretical properties.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-02-25T14:51:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "90C31",
      "62F35",
      "65K05",
      "94B35"
    ],
    "keywords": [
      "signal recovery",
      "regularization",
      "statistics",
      "fine error analysis",
      "classical linear regression problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1310.7637F"
    }
  }
}