Structured Symmetric Tensors

TL;DR

This paper introduces new classes of structured symmetric tensors, explores their properties, relations with classical cones, and applications in optimization, providing theoretical insights and answering open questions in tensor analysis.

Contribution

It defines and characterizes new tensor classes like CD, SSOS, and SOS$^*$, and studies their relations with classical cones, extending classical theorems and addressing open problems.

Findings

Complete Hankel tensors are CD tensors.

PSD Hankel tensor can be non-SOS, answering an open question.

Generalized Schur product theorem for CD and CP tensors.

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.