Fast truncation of mode ranks for bilinear tensor operations

D. V. Savostyanov, E. E. Tyrtyshnikov, N. L. Zamarashkin

Research output: Contribution to journalArticle

Abstract

We propose a fast algorithm for mode rank truncation of the result of a bilinear operation on 3-tensors given in the Tucker or canonical form. If the arguments and the result have mode sizes n and mode ranks r, the computation costs (nr 3 + r 4). The algorithm is based on the cross approximation of Gram matrices, and the accuracy of the resulted Tucker approximation is limited by square root of the machine precision. We apply the proposed algorithms for the evaluation of the Hadamard square of the electron density and demonstrate that it outperforms previously used methods for this purpose. We also check the accuracy of the resulted Tucker approximation and show that one iteration of the Tucker-ALS method improves it almost up to the machine precision.

Original languageEnglish
Pages (from-to)103-111
Number of pages9
JournalNumerical Linear Algebra with Applications
Volume19
Issue number1
DOIs
Publication statusPublished - 1 Jan 2012

Keywords

  • Cross approximation
  • Fast compression
  • Multidimensional arrays
  • Structured tensors
  • Tucker approximation

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    Savostyanov, D. V., Tyrtyshnikov, E. E., & Zamarashkin, N. L. (2012). Fast truncation of mode ranks for bilinear tensor operations. Numerical Linear Algebra with Applications, 19(1), 103-111. https://doi.org/10.1002/nla.765