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

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

Tensor decomposition and applications参考文献 52被引用 8

一句话总结

本文通过非负元素和H-特征值引入双非负张量，将矩阵概念扩展至任意阶高阶张量。证明了任意阶正柯西张量均为完全正定，并提出一种预处理方案，利用优势性质和改进的Fan-Zhou算法加速完全正定性的验证，应用于张量互补问题。

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

• 将双非负矩阵的概念扩展至高阶张量，包括正定性不再适用的奇数阶情形。
• 识别并表征完全正定张量的子类，特别是易于验证的子类。
• 开发一种预处理方案，高效排除在应用Fan-Zhou算法前非完全正定的张量。
• 在强双非负张量约束下，分析张量互补问题的解结构。
• 提供一个理论与计算框架，补充张量分析与优化，尤其针对奇数阶张量。

提出的方法

• 将双非负张量定义为具有非负元素和非负H-特征值的对称张量，强双非负张量则定义为具有正H-特征值的张量。
• 利用优势性质排除某些结构化张量（如m ≥ 3的m-一致超图的无符号拉普拉斯张量）为完全正定的可能性。
• 提出一种预处理的Fan-Zhou算法，利用优势检验与元素调整，加速完全正定性的验证。
• 通过非负系数的基张量的随机线性组合构造对称非负张量，以测试预处理效率。
• 证明所有任意阶（偶数或奇数）的正柯西张量均为完全正定，提供一种新的充分条件。
• 通过证明强双非负张量可保证张量互补问题（TCP）的解集非空且紧致，分析其解结构。

实验结果

研究问题

• RQ1 双非负性概念能否在正定性未定义的奇数阶张量中合理扩展？
• RQ2 在偶数阶情况下为正定的某些结构化张量，在奇数阶情况下是否仍保持双非负性？
• RQ3 基于优势的性质能否高效排除非完全正定张量？
• RQ4 是否存在一个广泛且易于验证的完全正定张量子类，包含奇数阶实例？
• RQ5 当张量为强双非负时，张量互补问题的解行为如何？

主要发现

• 任意阶（偶数或奇数）的正柯西张量均为完全正定，提供了一类新且易于构造的完全正定张量子类。
• 对于m ≥ 3的非空m-一致超图，其无符号拉普拉斯张量虽为双非负，但并非完全正定。
• 基于优势性质与元素调整的预处理方案可在完整验证前排除高达99.5%的随机生成对称非负张量。
• 当m ≥ 5且n ≥ 3时，预处理效率显著下降，m=12、n=4时排除率低至22.4%，表明高阶复杂性增加。
• 所提出的预处理Fan-Zhou算法通过早期过滤非完全正定张量，显著加速完全正定性验证。
• 强双非负张量可保证张量互补问题具有非空且紧致的解集，确保良好的理论性质。