Semidefinite relaxation algorithms of a partially symmetric CP-decomposition

报告人简介：国防科技大学教授，博士生导师。主要从事张量优化、电磁场计算理论与应用等研究。在国内外期刊发表学术论文60余篇。主持国家自然科学基金面上2项，973子专题1项，学校和军队各类科研若干项。获国家级教学成果二等奖1项，军队科技进步二等奖2项，山东省自然科学三等奖1项。多次到香港理工大学和新加坡南洋理工大学进行学术访问和交流。

报告内容：In this paper, we establish a equivalence relation between partially symmetric tensors and homogeneous polynomials, prove that every partially symmetric tensor has a partially symmetric 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. Since 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 and general partially symmetric CP-decomposition, etc. Numerical examples show that the algorithms are effective.