On decompositions and approximations of conjugate partial-symmetric tensors

Taoran Fu, Bo Jiang, Zhening Li

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Abstract

Hermitian matrices have played an important role in matrix theory and complex quadratic optimization. The high-order generalization of Hermitian matrices, conjugate partial-symmetric (CPS) tensors, have shown growing interest recently in tensor theory and computation, particularly in application-driven complex polynomial optimization problems. In this paper, we study CPS tensors with a focus on ranks, computing rank-one decompositions and approximations, as well as their applications. We prove constructively that any CPS tensor can be decomposed into a sum of rank-one CPS tensors, which provides an explicit method to compute such rank-one decompositions. Three types of ranks for CPS tensors are defined and shown to be different in general. This leads to the invalidity of the conjugate version of Comon's conjecture. We then study rank-one approximations and matricizations of CPS tensors. By carefully unfolding CPS tensors to Hermitian matrices, rank-one equivalence can be preserved. This enables us to develop new convex optimization models and algorithms to compute best rank-one approximations of CPS tensors. Numerical experiments from data sets in radar wave form design, elasticity tensor, and quantum entanglement are performed to justify the capability of our methods.
Original languageEnglish
Article number46
Number of pages37
JournalCalcolo
Volume58
Issue number4
Early online date10 Oct 2021
DOIs
Publication statusEarly online - 10 Oct 2021

Keywords

  • Conjugate partial-symmetric tensor
  • Hermitian matrix
  • Partial-symmetric tensor
  • Tensor rank
  • Rank-one decomposition
  • Rank-one approximation
  • Tensor eigenvalue
  • Tensor matricization

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