{
  "id": "2207.11141",
  "version": "v1",
  "published": "2022-07-22T15:25:59.000Z",
  "updated": "2022-07-22T15:25:59.000Z",
  "title": "Deep learning of diffeomorphisms for optimal reparametrizations of shapes",
  "authors": [
    "Elena Celledoni",
    "Helge Glöckner",
    "Jørgen Riseth",
    "Alexander Schmeding"
  ],
  "comment": "26 pages, 11 figures. Submitted to SIAM Journal of Scientific Computing",
  "categories": [
    "math.OC",
    "cs.LG",
    "math.DG"
  ],
  "abstract": "In shape analysis, one of the fundamental problems is to align curves or surfaces before computing a (geodesic) distance between these shapes. To find the optimal reparametrization realizing this alignment is a computationally demanding task which leads to an optimization problem on the diffeomorphism group. In this paper, we construct approximations of orientation-preserving diffeomorphisms by composition of elementary diffeomorphisms to solve the approximation problem. We propose a practical algorithm implemented in PyTorch which is applicable both to unparametrized curves and surfaces. We derive universal approximation results and obtain bounds for the Lipschitz constant of the obtained compositions of diffeomorphisms.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-07-22T15:25:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "65K10",
      "58D05",
      "46T10"
    ],
    "keywords": [
      "optimal reparametrization",
      "deep learning",
      "derive universal approximation results",
      "shape analysis",
      "lipschitz constant"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}