[论文解读] Sum of Squares Decompositions and Rank Bounds for Biquadratic Forms

Liqun Qi, Chunfeng Cui+1 more|arXiv (Cornell University)|Jan 8, 2026

Matrix Theory and Algorithms被引用 0

一句话总结

本文刻画双二次型的PSD与SOS性质，证明PSD x-对称形式总是SOS，并提供SOS分解的界限与利用Kronecker-积结构的有效计算方法，给出显式的SOS秩下界/上界。

ABSTRACT

We study SOS properties of biquadratic forms. 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 introduce simple biquadratic forms. For $m \ge 2$, we present a $m imes 2$ PSD biquadratic form and show that it can be expressed as the sum of $m+1$ squares, but cannot be expressed as the sum of $m$ squares. This provides a lower bound for sos rank of $m imes 2$ biquadratic forms, and shows that previously proved results that a $2 imes 2$ PSD biquadratic form can be expressed as the sum of three squares, and a $3 imes 2$ PSD biquadratic form can be expressed as the sum of four squares, are tight. We also present an $3 imes 3$ SOS biquadratic form, which can be expressed as the sum of six squares, but not the sum of five squares.We present a $2 imes 2$ 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 $3 imes 2$ 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 $2 imes 2$ PSD biquadratic form can be expressed as the sum of three squares, and a $3 imes 2$ PSD biquadratic form can be expressed as the sum of four squares, are tight. Moreover, we establish a universal upper bound SOS-rank$(P) \le mn-1$ for any SOS biquadratic form, which improves the trivial bound $mn$ and is tight in small dimensions.

研究动机与目标

• 研究正半定的双二次型及其SOS表示。
• 将已知的从完全对称扩展到部分对称（x-对称）形式的SOS结果。
• 开发一种利用Kronecker-积结构的计算过程以构造SOS分解。

提出的方法

• 定义单数与非单数的x-对称双二次型及其相关的对称双二次张量。
• 推导PSD的必要充分条件，通过矩阵不等式 I+B-A ≽ 0 与 I+B+(m−1)A ≽ 0。
• 证明每个PSD的单数x-对称双二次型都可通过从PSD的M矩阵构造的SOS分解来表示为SOS。
• 给出一个结构化的SOS秩界：SOS-rank(P) ≤ rank(R) + (m−1) rank(Q) 其中 Q = I_n + B − A 且 R = I_n + B + (m−1)A。
• 概述一个计算方法，利用结构化的 M = I_m ⊗ Q + (1/m)(1_m 1_m^T) ⊗ (R−Q) 及其特征分解来获得SOS分解。
• 在需要时通过降到单数情况并应用变量缩放，将结果推广到一般的x-对称形式。

实验结果

研究问题

• RQ1 在什么条件下PSD单数x-对称双二次型也是SOS？
• RQ2 PSD x-对称双二次型是否普遍存在SOS分解（不仅限于单数情况）？
• RQ3 小/一般维度下的SOS秩的紧界是什么？（m, n 的情况）
• RQ4 是否可以设计高效的计算过程，利用Kronecker-积结构来构造显式的SOS分解？

主要发现

• 一个PSD的单数x-对称双二次型始终是双线性形式的平方和。
• PSD的x-对称形式满足矩阵不等式 I+B−A ≽ 0 与 I+B+(m−1)A ≽ 0，且这两条为必要且充分条件。
• 单数x-对称形式的SOS-rank界为：SOS-rank(P) ≤ rank(R) + (m−1) rank(Q)，其中 Q = I_n + B − A 且 R = I_n + B + (m−1)A。
• 对于任意SOS双二次型，存在一个通用上界 SOS-rank(P) ≤ mn − 1，优于显性的 mn 界。
• 提供一个显式而高效的过程，利用Kronecker-积结构来计算SOS分解，得到的分解的秩等于 rank(M)。
• 下界构造表明在小维度下SOS-rank结果是紧的（例如 m×2 形式至少需要 m+1 个平方；一个具体的3×3例子需要六个平方）。