Sum of Squares Decompositions for Structured Biquadratic Forms

TL;DR

This paper investigates sum-of-squares representations for structured biquadratic forms, establishing conditions for positive semi-definiteness and SOS decompositions, and characterizing the geometry of the PSD cone.

Contribution

It proves that diagonally dominated symmetric biquadratic tensors are always SOS and characterizes the PSD cone for symmetric biquadratic forms, advancing understanding of SOS conditions.

Findings

Diagonally dominated symmetric biquadratic tensors are always SOS.

The PSD cone for symmetric biquadratic forms is a convex polyhedron.

Every symmetric biquadratic PSD form is SOS for any dimensions.

Abstract

This paper studies sum-of-squares (SOS) representations for structured biquadratic forms. We prove that diagonally dominated symmetric biquadratic tensors are always SOS. For the special case of symmetric biquadratic forms, we establish necessary and sufficient conditions for positive semi-definiteness of monic symmetric biquadratic forms, characterize the geometry of the corresponding PSD cone as a convex polyhedron, and prove that every such PSD form is SOS for any dimensions $m$ and $n$. We also formulate conjectures regarding SOS representations for symmetric M-biquadratic tensors and symmetric $\mathrm{B}_{0}$-biquadratic tensors, discussing their likelihood and potential proof strategies. Our results advance the understanding of when positive semi-definiteness implies sum-of-squares decompositions for structured biquadratic forms.