[论文解读] Doubly Nonnegative Tensors and Completely Positive Tensors

Ziyan Luo, Liqun Qi|arXiv (Cornell University)|Apr 29, 2015

Tensor decomposition and applications参考文献 24被引用 1

一句话总结

本文将双非负性从矩阵推广至任意阶张量，通过非负元素和非负H-特征值定义双非负张量。证明了所有正柯西张量均为完全非负，并提出一种预处理的Fan-Zhou算法以验证完全非负性，揭示了双非负张量锥与完全非负张量锥之间的结构性差异——尤其在奇数阶情况下，矩阵类比关系不再成立。

ABSTRACT

The concept of double nonnegativity of matrices is generalized to doubly nonnegative tensors by means of the nonnegativity of all entries and $H$-eigenvalues. This generalization is defined for tensors of any order (even or odd), while it reduces to the class of nonnegative positive semidefinite tensors in the even order case. We show that many nonnegative structured tensors, which are positive semidefinite in the even order case, are indeed doubly nonnegative as well in the odd order case. As an important subclass of doubly nonnegative tensors, the completely positive tensors are further studied. By using dominance properties for completely positive tensors, we can easily exclude some doubly nonnegative tensors, such as the signless Laplacian tensor of a nonempty $m$-uniform hypergraph with $m\geq 3$, from the class of completely positive tensors. Properties of the doubly nonnegative tensor cone and the completely positive tensor cone are established. Their relation and difference are discussed. These show us a different phenomenon comparing to the matrix case. By employing the proposed properties, more subclasses of these two types of tensors are identified. Particularly, all positive Cauchy tensors with any order are shown to be completely positive. This gives an easily constructible subclass of completely positive tensors, which is significant for the study of completely positive tensor decomposition. A preprocessed Fan-Zhou algorithm is proposed which can efficiently verify the complete positivity of nonnegative symmetric tensors. We also give the solution analysis of tensor complementarity problems with the strongly doubly nonnegative tensor structure.

研究动机与目标

• 将双非负性的概念从矩阵推广至任意阶张量，包括偶数阶与奇数阶。
• 表征双非负张量锥与完全非负张量锥之间的结构性差异，尤其在奇数阶情况下。
• 识别完全非负张量的新子类，特别是证明所有正柯西张量均为完全非负。
• 开发一种高效算法以验证非负对称张量的完全非负性。
• 在强双非负张量结构下分析张量互补问题。

提出的方法

• 通过要求所有元素和H-特征值均为非负，将双非负性推广至张量。
• 引入支配性质以排除某些双非负张量（如m阶均匀超图的无符号拉普拉斯张量，其中m ≥ 3）成为完全非负张量的可能性。
• 利用谱性质与张量分解理论，建立双非负张量锥与完全非负张量锥之间的关系。
• 提出Fan-Zhou算法的预处理版本，以高效验证非负对称张量的完全非负性。
• 将张量互补问题框架应用于强双非负张量结构下的解分析。
• 以正柯西张量结构作为典型示例，展示完全非负张量子类的可构造性。

实验结果

研究问题

• RQ1 双非负性概念如何从矩阵推广至任意阶张量？
• RQ2 在奇数阶情况下，双非负张量锥与完全非负张量锥之间存在何种结构性差异？
• RQ3 哪些非负张量类可被证明为完全非负，尤其是超出偶数阶半正定情形？
• RQ4 能否设计一种高效算法以验证对称非负张量的完全非负性？
• RQ5 在强双非负张量结构下，张量互补问题的行为如何？

主要发现

• 所有任意阶的正柯西张量均为完全非负，提供了一类易于构造的完全非负张量子类。
• 当m ≥ 3时，非空m阶均匀超图的无符号拉普拉斯张量为双非负但非完全非负，揭示了与矩阵情形的关键差异。
• 在奇数阶张量中，双非负张量锥与完全非负张量锥存在显著差异，而这一差异在矩阵情形下于特定条件下并不存在。
• 预处理的Fan-Zhou算法可高效验证非负对称张量的完全非负性。
• 具有强双非负张量结构的张量互补问题可进行明确定义的解分析，扩展了矩阵情形下的已知结果。
• 支配性质提供了一种实用工具，用于排除某些双非负张量成为完全非负张量的可能性，从而增强结构性表征。