[Paper Review] MB-tensors and MB0-tensors

Chaoqian Li, Yaotang Li|arXiv (Cornell University)|Nov 24, 2014

Tensor decomposition and applications15 references4 citations

TL;DR

This paper introduces MB-tensors and MB0-tensors as a generalization of B-tensors and quasi-double B-tensors, proving that even-order symmetric MB-tensors are positive definite and even-order symmetric MB0-tensors are positive semi-definite—thereby confirming a conjecture by Li and Li regarding quasi-double B0-tensors.

ABSTRACT

The class of MB(MB0)-tensors, which is a generation of B(B0)-tensors and quasi-double B(B0)-tensors, is proposed. And we prove that an even order symmetric MB(MB0)-tensor is positive (semi-)definite. This provides a positive answer for the conjecture in Li and Li's paper [15] that an even order symmetric quasi-double B0-tensor is positive semi-definite.

Motivation & Objective

• To propose a new class of tensors, MB-tensors and MB0-tensors, as a generalization of B-tensors and quasi-double B-tensors.
• To establish the positive (semi-)definiteness of even-order symmetric MB(MB0)-tensors.
• To provide a positive answer to the conjecture in Li and Li's paper [15] regarding the positive semi-definiteness of even-order symmetric quasi-double B0-tensors.

Proposed method

• The paper defines MB-tensors and MB0-tensors through a generalization of the B-tensor structure, incorporating additional constraints on diagonal dominance.
• It introduces a recursive construction method based on the comparison of diagonal entries and off-diagonal entries in the tensor structure.
• The proof of positive definiteness relies on analyzing the sign of multilinear forms associated with even-order symmetric tensors.
• It leverages known results on B-tensors and quasi-double B-tensors as foundational components in the generalization.
• The authors use properties of nonnegative tensors and diagonal dominance to establish the positive definiteness of MB(MB0)-tensors.
• The analysis is restricted to even-order symmetric tensors, where structural symmetry enables the derivation of definiteness properties.

Experimental results

Research questions

• RQ1 Is the class of MB-tensors a valid generalization of B-tensors and quasi-double B-tensors?
• RQ2 Are even-order symmetric MB-tensors positive definite?
• RQ3 Are even-order symmetric MB0-tensors positive semi-definite?
• RQ4 Does the proposed class of tensors resolve the conjecture on quasi-double B0-tensors?
• RQ5 What structural properties ensure positive (semi-)definiteness in this generalized tensor class?

Key findings

• The class of MB-tensors is introduced as a proper generalization of B-tensors and quasi-double B-tensors.
• Every even-order symmetric MB-tensor is proven to be positive definite.
• Every even-order symmetric MB0-tensor is proven to be positive semi-definite.
• The conjecture by Li and Li [15] regarding the positive semi-definiteness of even-order symmetric quasi-double B0-tensors is confirmed to be true.
• The structural properties of MB(MB0)-tensors ensure that their associated multilinear forms are non-negative (positive) for all non-zero vectors.