A Semidefinite Relaxation Method for Partially Symmetric Tensor Decomposition

• 成果类型: Article; Early Access
• 署名作者: Ni, Guyan; Li, Ying
• 署名单位: National University of Defense Technology - China
• 刊物名称: MATHEMATICS OF OPERATIONS RESEARCH
• ISSN/ISSBN: 0364-765X
• DOI: 10.1287/moor.2021.1231
• 发表日期: 2022
• 关键词: canonical decomposition; rank; approximation; optimization; POLYNOMIALS

摘要

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,mi,ni, 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.