55
views
0
recommends
+1 Recommend
0 collections
    0
    shares
      • Record: found
      • Abstract: found
      • Article: found
      Is Open Access

      Regularized Computation of Approximate Pseudoinverse of Large Matrices Using Low-Rank Tensor Train Decompositions

      Preprint
      ,

      Read this article at

      Bookmark
          There is no author summary for this article yet. Authors can add summaries to their articles on ScienceOpen to make them more accessible to a non-specialist audience.

          Abstract

          We propose a new method for low-rank approximation of Moore-Penrose pseudoinverses (MPPs) of large-scale matrices using tensor networks. The computed pseudoinverses can be useful for solving or preconditioning of large-scale overdetermined or underdetermined systems of linear equations. The computation is performed efficiently and stably based on the modified alternating least squares (MALS) scheme using low-rank tensor train (TT) decompositions and tensor network contractions. The formulated large-scale optimization problem is reduced to sequential smaller-scale problems for which any standard and stable algorithms can be applied. Regularization technique is incorporated in order to alleviate ill-posedness and obtain robust low-rank approximations. Numerical simulation results illustrate that the regularized pseudoinverses of a wide class of non-square or nonsymmetric matrices admit good approximate low-rank TT representations. Moreover, we demonstrated that the computational cost of the proposed method is only logarithmic in the matrix size given that the TT-ranks of a data matrix and its approximate pseudoinverse are bounded. It is illustrated that a strongly nonsymmetric convection-diffusion problem can be efficiently solved by using the preconditioners computed by the proposed method.

          Related collections

          Author and article information

          Journal
          2015-06-05
          2015-12-22
          Article
          1506.01959
          28fead2c-59e1-47ed-8041-c8c53c1032c8

          http://arxiv.org/licenses/nonexclusive-distrib/1.0/

          History
          Custom metadata
          15A09, 65F08, 65F20, 65F22
          28 pages
          math.NA cs.NA

          Numerical & Computational mathematics
          Numerical & Computational mathematics

          Comments

          Comment on this article