Completely Positive Tensors: Properties, Easily Checkable Subclasses, and Tractable Relaxations

Abstract

The completely positive (CP) tensor verification and decomposition are essential in tensor analysis and computation due to the wide applications in statistics, computer vision, exploratory multiway data analysis, blind source separation and polynomial optimization. However, it is generally NP-hard as we know from its matrix case. To facilitate the CP tensor verification and decomposition, more properties for the CP tensor are further studied, and a great variety of its easily checkable subclasses such as the positive Cauchy tensors, the symmetric Pascal tensors, the Lehmer tensors, the power mean tensors, and all of their nonnegative fractional Hadamard powers and Hadamard products, are exploited in this paper. Particularly, a so-called CP-Vandermonde decomposition for positive Cauchy-Hankel tensors is established and a numerical algorithm is proposed to obtain such a special type of CP decomposition. The doubly nonnegative (DNN) matrix is generalized to higher order tensors as well. Based on the DNN tensors, a series of tractable outer approximations are characterized to approximate the CP tensor cone, which serve as potential useful surrogates in the corresponding CP tensor cone programming arising from polynomial programming problems.

A semidefinite relaxation algorithm for checking completely positive separable matrices

“…It is easy to check that A is doubly nonnegative with the order 4, so A ∈ CP 4 = CP 1,4 = CP 4,1 . Indeed, all Pascal matrices A ∈ S n are completely positive matrices [18]. (i) Firstly, we verify that A ∈ CP 4,1 by Algorithm 4.1.…”

“…It was shown in [18] that all Lehmer matrices are completely positive. Here, we consider the case n = 6.…”

Inequalities on Generalized Tensor Functions with Diagonalizable and Symmetric Positive Definite Tensors

“…For a given b) if D is positive semi-definite with all diagonal entries nonzero, and B is nonsingular, then A is positive definite and is called a completely decomposable tensor [17]; c) if D is positive semi-definite with all diagonal entries nonnegative, and B is nonnegative, then A is called a completely positive tensor. For the properties and checkability of completely positive tensors, the interested readers are referred to [11,19,26,30,31] and the references therein.…”

A hierarchy of semidefinite relaxations for completely positive tensor optimization problems

“…The CP tensor is a natural extension of the CP matrix. We refer to [1,2,21,35,36] for work on CP matrices and tensors. For B ∈ S m (R n ), we define…”