[Paper Review] Properties of Some Classes of Structured Tensors

Yisheng Song, Liqun Qi|arXiv (Cornell University)|Mar 5, 2014

Tensor decomposition and applications34 references4 citations

TL;DR

This paper extends P, P₀, B, and B₀ matrix concepts to higher-order tensors, establishing that symmetric P (P₀) tensors are equivalent to positive (semi-)definite tensors. It proves that principal sub-tensors of such tensors retain their class membership and links these tensors to optimization, nonlinear complementarity problems, and nonnegative tensor theory, providing foundational insights for future algorithmic and theoretical development in tensor analysis.

ABSTRACT

In this paper, we extend some classes of structured matrices to higher order tensors. We discuss their relationships with positive semi-definite tensors and some other structured tensors. We show that every principal sub-tensor of such a structured tensor is still a structured tensor in the same class, with a lower dimension. The potential links of such structured tensors with optimization, nonlinear equations, nonlinear complementarity problems, variational inequalities and the nonnegative tensor theory are also discussed.

Motivation & Objective

• To extend the theory of structured matrices (P, P₀, B, B₀) to higher-order tensors.
• To investigate the relationship between P/P₀ tensors and positive (semi-)definite tensors.
• To explore the structural properties of these tensors, particularly under principal sub-tensor operations.
• To establish connections with optimization, nonlinear complementarity problems, and nonnegative tensor theory.
• To identify open problems and guide future research in tensor analysis and applications.

Proposed method

• Define P and P₀ tensors via max and non-negative conditions on x_i (Ax)_i for nonzero x.
• Extend eigenvalue concepts (H-, Z-eigenvalues) to general tensors and analyze their behavior under symmetry and odd-order constraints.
• Prove that every principal sub-tensor of a P (P₀) tensor remains in the same class.
• Introduce B and B₀ tensors as extensions of B matrices, showing that a Z-tensor is diagonally dominated if and only if it is a B₀ tensor.
• Use spectral theory and tensor decomposition to analyze positivity and definiteness in symmetric and non-symmetric settings.
• Link the tensor classes to optimization problems, nonlinear equations, and variational inequalities through structural and spectral properties.

Experimental results

Research questions

• RQ1 Can P and P₀ matrices be naturally extended to higher-order tensors, and if so, how do they relate to positive (semi-)definite tensors?
• RQ2 Is every symmetric P (P₀) tensor equivalent to a positive (semi-)definite tensor, and what are the implications for odd-order tensors?
• RQ3 Do odd-order symmetric P tensors exist, and what are their spectral properties, particularly regarding Z-eigenvalues?
• RQ4 Are B and B₀ tensors diagonally dominated, and how do they relate to M-tensors and Laplacian tensors?
• RQ5 What is the relationship between non-negative B (B₀) tensors and completely positive tensors or positive semi-definite tensors?

Key findings

• A symmetric tensor is a P (P₀) tensor if and only if it is positive (semi-)definite, establishing a direct equivalence between the two concepts.
• Every principal sub-tensor of a P (P₀) tensor is also a P (P₀) tensor of lower dimension, preserving structural integrity.
• There does not exist an odd-order symmetric P tensor, and any odd-order non-symmetric P tensor has no Z-eigenvalues.
• An odd-order P₀ tensor has no nonzero Z-eigenvalues, indicating strong spectral restrictions.
• A Z-tensor is diagonally dominated if and only if it is a B₀ tensor, providing a checkable condition for this class.
• The paper identifies open questions and future research directions, including spectral algorithms and connections to nonnegative tensor theory.