A Semidefinite Relaxation Method for Partially Symmetric Tensor Decomposition

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 $\mathcal{T}\in S[\mathbf{m}]\mathbb{F}[\mathbf{n}]$ 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.