A Semidefinite Relaxation Method for Partially Symmetric Tensor Decomposition

Guyan Ni () and Ying Li ()
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Guyan Ni: Department of Mathematics, National University of Defense Technology, Changsha, Hunan 410073, China
Ying Li: Department of Mathematics, National University of Defense Technology, Changsha, Hunan 410073, China

Mathematics of Operations Research, 2022, vol. 47, issue 4, 2931-2949

Abstract: In this paper, we establish an equivalence relation between partially symmetric tensors and homogeneous polynomials, prove that every partially symmetric tensor has a partially symmetric canonical polyadic (CP)-decomposition, and present three semidefinite relaxation algorithms. The first algorithm is used to check whether there exists a positive partially symmetric real CP-decomposition for a partially symmetric real tensor and give a decomposition if it has. The second algorithm is used to compute general partial symmetric real CP-decompositions. The third algorithm is used to compute positive partially symmetric complex CP-decomposition of partially symmetric complex tensors. Because for different parameters s , m i , n i , partially symmetric tensors T ∈ S [ m ] F [ n ] represent different kinds of tensors. Hence, the proposed algorithms can be used to compute different types of tensor real/complex CP-decomposition, including general nonsymmetric CP-decomposition, positive symmetric CP-decomposition, positive partially symmetric CP-decomposition, general partially symmetric CP-decomposition, etc. Numerical examples show that the algorithms are effective.

Keywords: Primary: 15A18; 15A69; 41A65; secondary: 14P10; 90C22; partially symmetric tensor; semidefinite relaxation; tensor CP-decomposition; truncated moment problem; polynomial optimization (search for similar items in EconPapers)
Date: 2022
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