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Sum of Squares Decompositions and Rank Bounds for Biquadratic Forms

Liqun Qi, Chunfeng Cui, Yi Yu

TL;DR

The paper resolves the SOS question for partially symmetric biquadratic forms by showing that every PSD $x$-symmetric (and by symmetry, $y$-symmetric) form is SOS, and it derives a tight bound on the SOS rank using a Kronecker-product representation. It provides necessary and sufficient PSD conditions, a constructive SOS decomposition, and an explicit, structure-aware bound on the SOS rank, together with an efficient algorithm based on the Kronecker structure that reduces computational complexity. Tightness results for $2\times 2$ and $3\times 2$ forms demonstrate that previously known bounds are optimal. Open questions include characterizing maximal SOS rank across general $m,n$, and extending the approach to other symmetry types and higher-degree multiquadratic forms, with practical impact for certifying nonnegativity in optimization and applied mathematics.

Abstract

We study positive semi-definite (PSD) biquadratic forms and their sum-of-squares (SOS) representations. For the class of partially symmetric biquadratic forms, we establish necessary and sufficient conditions for positive semi-definiteness and prove that every PSD partially symmetric biquadratic form is a sum of squares of bilinear forms. This extends the known result for fully symmetric biquadratic forms. We describe an efficient computational procedure for constructing SOS decompositions, exploiting the Kronecker-product structure of the associated matrix representation. We present a $2 \times 2$ PSD biquadratic form, and show that it can be expressed as the sum of three squares, but cannot be expressed as the sum of two squares. Furthermore, we present a $3 \times 2$ PSD biquadratic form, and show that it can be expressed as the sum of four squares, but cannot be expressed as the sum of three squares. These show that previously proved results that a $2 \times 2$ PSD biquadratic form can be expressed as the sum of three squares, and a $3 \times 2$ PSD biquadratic form can be expressed as the sum of four squares, are tight.

Sum of Squares Decompositions and Rank Bounds for Biquadratic Forms

TL;DR

The paper resolves the SOS question for partially symmetric biquadratic forms by showing that every PSD -symmetric (and by symmetry, -symmetric) form is SOS, and it derives a tight bound on the SOS rank using a Kronecker-product representation. It provides necessary and sufficient PSD conditions, a constructive SOS decomposition, and an explicit, structure-aware bound on the SOS rank, together with an efficient algorithm based on the Kronecker structure that reduces computational complexity. Tightness results for and forms demonstrate that previously known bounds are optimal. Open questions include characterizing maximal SOS rank across general , and extending the approach to other symmetry types and higher-degree multiquadratic forms, with practical impact for certifying nonnegativity in optimization and applied mathematics.

Abstract

We study positive semi-definite (PSD) biquadratic forms and their sum-of-squares (SOS) representations. For the class of partially symmetric biquadratic forms, we establish necessary and sufficient conditions for positive semi-definiteness and prove that every PSD partially symmetric biquadratic form is a sum of squares of bilinear forms. This extends the known result for fully symmetric biquadratic forms. We describe an efficient computational procedure for constructing SOS decompositions, exploiting the Kronecker-product structure of the associated matrix representation. We present a PSD biquadratic form, and show that it can be expressed as the sum of three squares, but cannot be expressed as the sum of two squares. Furthermore, we present a PSD biquadratic form, and show that it can be expressed as the sum of four squares, but cannot be expressed as the sum of three squares. These show that previously proved results that a PSD biquadratic form can be expressed as the sum of three squares, and a PSD biquadratic form can be expressed as the sum of four squares, are tight.
Paper Structure (9 sections, 6 theorems, 61 equations)

This paper contains 9 sections, 6 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Partially Symmetric Biquadratic Forms
  3. Positive Semi-definiteness of Monic $x$-Symmetric Forms
  4. SOS Property of Monic $x$-Symmetric Forms
  5. Bounding the SOS Rank
  6. Extension to General $x$-Symmetric Forms
  7. Computational Method for SOS Decomposition
  8. Tigntnesss of the SOS Rank Bounds for $2 \times 2$ and $3 \times 2$ Forms
  9. Concluding Remarks

Key Result

Theorem 2.2

Let $P$ be an $m \times n$ monic $x$-symmetric biquadratic form as in equ:monic_sbq, and let $\mathcal{A}\in SBQ(m,n)$ be the corresponding symmetric biquadratic tensor. Let $A=(a_{jl})$ and $B=(b_{jl})$ be the $n\times n$ symmetric matrices of coefficients. Then $P$ is PSD if and only if the follow where $I$ denotes the $n \times n$ identity matrix.

Theorems & Definitions (19)

  • Definition 2.1
  • Theorem 2.2
  • proof
  • Remark 2.3
  • Theorem 2.4
  • proof
  • Remark 2.5
  • Theorem 2.6: SOS rank bound
  • proof
  • Remark 2.7
  • ...and 9 more