Structured Symmetric Tensors
Liqun Qi, Chunfeng Cui, Yi Xu
TL;DR
This work develops a comprehensive framework for structured symmetric tensors by introducing three new classes—completely decomposable (CD) tensors, strictly sum-of-squares (SSOS) tensors, and SOS$^*$ tensors—and positioning them within the landscape of classical cones $PSD_{m,n}$, $SOS_{m,n}$, $CP_{m,n}$, and $COP_{m,n}$. It establishes dualities, interior characterizations, and extensions of the Schur product theorem to CD and CP tensors, and proves an equivalence between strongly completely decomposable (SCD) and positive definiteness (PD). The paper further connects complete Hankel tensors to CD structure, demonstrates inheritance properties under Hankel constructions, and provides explicit examples of PSD Hankel tensors that are not SOS or CD, thereby clarifying strict inclusions. The SOS$^*$ cone is characterized via moment matrices and polynomial nonnegativity, and its relationship to the CD and SOS cones is clarified through duality and special-case equalities, with implications for semidefinite programming, moment problems, and robust polynomial optimization. Overall, the results offer new structural insights, duality-based tools, and practical implications for polynomial and tensor optimization, data analysis, and signal processing.
Abstract
In this paper, we study structured symmetric tensors. We introduce several new classes of structured symmetric tensors: completely decomposable (CD) tensors, strictly sum of squares (SSOS) tensors and SOS$^*$ tensors. CD tensors have applications in data analysis and signal processing. Complete Hankel tensors are CD tensors. SSOS tensors are defined as SOS tensors with a positive definite Gram matrix, ensuring structural stability under perturbations. The SOS$^*$ cone is defined as the dual cone of the SOS tensor cone, with characterizations via moment matrices and polynomial nonnegativity. We study the relations among completely positive (CP) cones, CD cones, sum of squares (SOS) cones, positive semidefinite (PSD) cones and copositive (COP) cones. We identify the interiors of PSD, SOS, CP, COP and CD cones for even-order tensors. These characterizations are crucial for interior-point methods and stability analysis in polynomial and tensor optimization. We generalize the classical Schur product theorem to CD and CP tensors, including the case of strongly completely decomposable (SCD) and strongly completely positive (SCP) tensors. We identify equivalence between strictly CD (SCD) and positive definite (PD) for CD tensors. Furthermore, we give an example of a PSD but not SOS Hankel tensor. This answers an open question raised in the literature.