Inheritance properties and sum-of-squares decomposition of Hankel tensors: theory and algorithms

Weiyang Ding, Liqun Qi, Yimin Wei

Research output: Journal article publicationJournal articleAcademic researchpeer-review

12 Citations (Scopus)

Abstract

� 2016, Springer Science+Business Media Dordrecht. In this paper, we show that if a lower-order Hankel tensor is positive semi-definite (or positive definite, or negative semi-definite, or negative definite, or SOS), then its associated higher-order Hankel tensor with the same generating vector, where the higher order is a multiple of the lower order, is also positive semi-definite (or positive definite, or negative semi-definite, or negative definite, or SOS, respectively). Furthermore, in this case, the extremal H-eigenvalues of the higher order tensor are bounded by the extremal H-eigenvalues of the lower order tensor, multiplied with some constants. Based on this inheritance property, we give a concrete sum-of-squares decomposition for each strong Hankel tensor. Then we prove the second inheritance property of Hankel tensors, i.e., a Hankel tensor has no negative (or non-positive, or positive, or nonnegative) H-eigenvalues if the associated Hankel matrix of that Hankel tensor has no negative (or non-positive, or positive, or nonnegative, respectively) eigenvalues. In this case, the extremal H-eigenvalues of the Hankel tensor are also bounded by the extremal eigenvalues of the associated Hankel matrix, multiplied with some constants. The third inheritance property of Hankel tensors is raised as a conjecture.
Original languageEnglish
Pages (from-to)169-190
Number of pages22
JournalBIT Numerical Mathematics
Volume57
Issue number1
DOIs
Publication statusPublished - 1 Mar 2017

Keywords

  • Convolution
  • Hankel tensor
  • Inheritance property
  • Positive semi-definite tensor
  • Sum-of-squares

ASJC Scopus subject areas

  • Software
  • Computer Networks and Communications
  • Computational Mathematics
  • Applied Mathematics

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