{
  "id": "1908.03006",
  "version": "v1",
  "published": "2019-08-08T10:38:01.000Z",
  "updated": "2019-08-08T10:38:01.000Z",
  "title": "Sparse $\\ell^q$-regularization of inverse problems with deep learning",
  "authors": [
    "Markus Haltmeier",
    "Linh Nguyen",
    "Daniel Obmann",
    "Johannes Schwab"
  ],
  "categories": [
    "math.NA",
    "cs.LG",
    "cs.NA",
    "math.OC"
  ],
  "abstract": "We propose a sparse reconstruction framework for solving inverse problems. Opposed to existing sparse reconstruction techniques that are based on linear sparsifying transforms, we train an encoder-decoder network $D \\circ E$ with $E$ acting as a nonlinear sparsifying transform. We minimize a Tikhonov functional which used a learned regularization term formed by the $\\ell^q$-norm of the encoder coefficients and a penalty for the distance to the data manifold. For this augmented sparse $\\ell^q$-approach, we present a full convergence analysis, derive convergence rates and describe a training strategy. As a main ingredient for the analysis we establish the coercivity of the augmented regularization term.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-08-08T10:38:01.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "inverse problems",
      "deep learning",
      "existing sparse reconstruction techniques",
      "full convergence analysis",
      "sparse reconstruction framework"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}