Completely Positive Biquadratic Tensors

TL;DR

Addresses the structure of completely positive biquadratic tensors and their copositive counterparts, introducing $CPBQ(m,n)$ and $COPSBQ(m,n)$ and establishing duality between the cones. Establishes that all weakly CPB tensors are SOS biquadratic and that decomposable CPB tensors arise from outer products of CP matrices. Analyzes two easily checkable subclasses, positive biquadratic Cauchy tensors and biquadratic Pascal tensors, proving that Pascal tensors are positive definite and strongly CPB, with integral representations and finite CP decompositions to certify CPB. These results offer concrete criteria and constructions for CPB tensors with implications for optimization and hypergraph theory.

Abstract

In this paper, we systemically introduce completely positive biquadratic (CPB) tensors and copositive biquadratic tensors. We show that all weakly CPB tensors are sum of squares tensors, the CPB tensor cone and the copositive biquadratic tensor cone are dual cone to each other. We also show that the outer product of two completely positive matrices is a CPB tensor, and the outer product of two copositive matrices is a copositive biquadratic tensor. We then study two easily checkable subclasses of CPB tensors, namely positive biquadratic Cauchy tensors and biquadratic Pascal tensors. We show that a biquadratic Pascal tensor is both strongly CPB and positive definite.

Theorems & Definitions (12)

• Theorem 2.1
• proof
• Theorem 2.2
• proof
• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Theorem 4.1
• proof