Doubly Nonnegative Tensors, Completely Positive Tensors and Applications

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 ≥ 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.

{0,1} completely positive tensors and multi-hypergraphs

“…The aforementioned property is closely related to the zero-entry dominance property for tensors described formally by Luo and Qi in [21].…”

“…For the verification of cp tensors, an efficient approach in terms of truncated moment sequences for checking completely positive tensors was proposed and an optimization algorithm based on semidefinite relaxation for completely positive tensor decomposition was established by Fan and Zhou in [14]. This approach was later accelerated with some preprocessing steps by Luo and Qi in [21]. Some structured and geometrical properties on general cp tensors were also discussed in [21,26].…”

“…This approach was later accelerated with some preprocessing steps by Luo and Qi in [21]. Some structured and geometrical properties on general cp tensors were also discussed in [21,26].…”

“…It is known from [21] that any completely positive tensor possesses the zero-entry dominance property. It is worth pointing out that the zero-entry dominance property is only a necessary condition for (0, 1) associated tensor to be {0, 1} cp tensors, but far away from sufficient, even for the matrix case.…”