{
  "id": "1410.6089",
  "version": "v1",
  "published": "2014-10-22T16:02:54.000Z",
  "updated": "2014-10-22T16:02:54.000Z",
  "title": "Low-rank approximation of tensors",
  "authors": [
    "Shmuel Friedland",
    "Venu Tammali"
  ],
  "comment": "28 pages, 5 figures",
  "categories": [
    "math.NA"
  ],
  "abstract": "In many applications such as data compression, imaging or genomic data analysis, it is important to approximate a given tensor by a tensor that is sparsely representable. For matrices, i.e. 2-tensors, such a representation can be obtained via the singular value decomposition, which allows to compute best rank k-approximations. For very big matrices a low rank approximation using SVD is not computationally feasible. In this case different approximations are available. It seems that variants of CUR decomposition are most suitable. For d-mode tensors T with d>2, many generalizations of the singular value decomposition have been proposed to obtain low tensor rank decompositions. The most appropriate approximation seems to be best (r_1,...,r_d)-approximation, which maximizes the l_2 norm of the projection of T on a tensor product of subspaces U_1,...,U_d, where U_i is an r_i-dimensional subspace. One of the most common method is the alternating maximization method (AMM). It is obtained by maximizing on one subspace U_i, while keeping all other fixed, and alternating the procedure repeatedly for i=1,...,d. Usually, AMM will converge to a local best approximation. This approximation is a fixed point of a corresponding map on Grassmannians. We suggest a Newton method for finding the corresponding fixed point. We also discuss variants of CUR-approximation method for tensors. The first part of the paper is a survey on low rank approximation of tensors. The second new part of this paper is a new Newton method for best $(r_1,...,r_d)$-approximation. We compare numerically different approximation methods.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-10-22T16:02:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "15A69",
      "58C30",
      "65C05",
      "65F30"
    ],
    "keywords": [
      "low-rank approximation",
      "low rank approximation",
      "singular value decomposition",
      "low tensor rank decompositions",
      "newton method"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1410.6089F"
    }
  }
}