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Strongly Positive Semi-Definite Tensors and Strongly SOS Tensors

Liqun Qi, Chunfeng Cui

TL;DR

The paper extends PSD concepts to odd-order tensors by introducing strongly PSD and strongly SOS tensors, and distinguishes genuinely PSD from barren tensors via $H$-eigenvalues of the symmetrization. It shows SPSD implies genuinely PSD, studies dual cones and Hadamard-product behavior, and situates these notions among familiar structured tensors such as symmetric $M$-tensors, Laplacian tensors, and completely positive tensors. It further connects strong SOS to SOS properties, proving that completely positive tensors are strongly SOS and that strict Hankel tensors (including odd-order Hilbert tensors) are strongly SOS. By introducing barren tensors and analyzing generalized PSD, the work broadens the PSD-like landscape for odd-order tensors with implications for spectral hypergraph theory and tensor optimization, while outlining several open questions for future research.

Abstract

We introduce {odd-order} strongly PSD (positive semi-definite) tensors which map real vectors to nonnegative vectors. We then introduce odd-order strongly SOS (sum-of-squares) tensors. A strongly SOS tensor maps real vectors to nonnegative vectors whose components are all SOS polynomials. Strongly SOS tensors are strongly PSD tensors. Odd order completely positive tensors are strongly SOS tensors. We also introduce strict Hankel tensors, which are also strongly SOS tensors. Odd order Hilbert tensors are strict Hankel tensors. However, the Laplacian tensor of a uniform hypergraph may not be strongly PSD. This motivates us to study wider PSD-like tensors. A cubic tensor is said to be a generalized PSD tensor if its corresponding symmetrization tensor has no negative H-eigenvalue. In the odd order case, this extension contains a peculiar tensor class, whose members have no H-eigenvalues at all. We call such tensors barren tensors, and the other generalized PSD symmetric tensors genuinely PSD symmetric tensors. In the odd order case, genuinely PSD tensors embrace various useful structured tensors, such as strongly PSD tensors, symmetric M-tensors and Laplacian tensors of uniform hypergraphs. However, there are exceptions. It is known an even order symmetric B-tensor is a PSD tensor. We give an example of an odd order symmetric B-tensor, which is a barren tensor.

Strongly Positive Semi-Definite Tensors and Strongly SOS Tensors

TL;DR

The paper extends PSD concepts to odd-order tensors by introducing strongly PSD and strongly SOS tensors, and distinguishes genuinely PSD from barren tensors via -eigenvalues of the symmetrization. It shows SPSD implies genuinely PSD, studies dual cones and Hadamard-product behavior, and situates these notions among familiar structured tensors such as symmetric -tensors, Laplacian tensors, and completely positive tensors. It further connects strong SOS to SOS properties, proving that completely positive tensors are strongly SOS and that strict Hankel tensors (including odd-order Hilbert tensors) are strongly SOS. By introducing barren tensors and analyzing generalized PSD, the work broadens the PSD-like landscape for odd-order tensors with implications for spectral hypergraph theory and tensor optimization, while outlining several open questions for future research.

Abstract

We introduce {odd-order} strongly PSD (positive semi-definite) tensors which map real vectors to nonnegative vectors. We then introduce odd-order strongly SOS (sum-of-squares) tensors. A strongly SOS tensor maps real vectors to nonnegative vectors whose components are all SOS polynomials. Strongly SOS tensors are strongly PSD tensors. Odd order completely positive tensors are strongly SOS tensors. We also introduce strict Hankel tensors, which are also strongly SOS tensors. Odd order Hilbert tensors are strict Hankel tensors. However, the Laplacian tensor of a uniform hypergraph may not be strongly PSD. This motivates us to study wider PSD-like tensors. A cubic tensor is said to be a generalized PSD tensor if its corresponding symmetrization tensor has no negative H-eigenvalue. In the odd order case, this extension contains a peculiar tensor class, whose members have no H-eigenvalues at all. We call such tensors barren tensors, and the other generalized PSD symmetric tensors genuinely PSD symmetric tensors. In the odd order case, genuinely PSD tensors embrace various useful structured tensors, such as strongly PSD tensors, symmetric M-tensors and Laplacian tensors of uniform hypergraphs. However, there are exceptions. It is known an even order symmetric B-tensor is a PSD tensor. We give an example of an odd order symmetric B-tensor, which is a barren tensor.
Paper Structure (5 sections, 14 theorems, 28 equations, 2 figures)

This paper contains 5 sections, 14 theorems, 28 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Genuinely Positive Semi-Definite Tensors and Barren Tensors
  3. Strongly Positive Semi-Definite Tensors
  4. Strongly SOS Tensors and Strict Hankel Tensors
  5. Final Remarks

Key Result

Proposition 2.1

If $\mathcal{A} \in T_{m, n}$ is a nonzero genuinely positive definite tensor, then $-\mathcal{A}$ is not a genuinely positive definite tensor. On the other hand, if $\mathcal{A} \in T_{m, n}$ is a barren tensor, then $-\mathcal{A}$ is also a barren tensor.

Figures (2)

  • Figure 1: The relationship among generalized positive semi-definite tensors, genuinely positive semi-definite tensors, barren tensors, strongly positive semi-definite tensors, and completely positive tensors for both even and odd order cases. Here, PSD is the abbreviation for positive semi-definite.
  • Figure 2: The relations among strongly positive semi-definite tensors, strongly sum of squares tensors, completely positive tensors, and strict Hankel tensors for odd order tensors.

Theorems & Definitions (40)

  • Definition 1.1
  • Definition 1.2
  • Proposition 2.1
  • proof
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Example 2.5
  • Example 2.6
  • Example 2.7
  • ...and 30 more