A semidefinite algorithm for completely positive tensor decomposition

Abstract

A symmetric tensor, which has a symmetric nonnegative decomposition, is called a completely positive tensor. We consider the completely positive tensor decomposition problem. A semidefinite algorithm is presented for checking whether a symmetric tensor is completely positive. If it is not completely positive, a certificate for it can be obtained; if it is completely positive, a nonnegative decomposition can be obtained.2000 Mathematics Subject Classification. Primary 15A18, 15A69, 90C22.

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.…”

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

“…Obviously, these two cones are dual to each other and are both actually closed, convex, pointed, and full-dimensional. To test the membership of CP m,n , Fan and Zhou [20] has provided an optimization algorithm based on semidefinite relaxation. Besides the Fan-Zhou's algorithmic verification, properties for CP-tensors can sometimes help us to verify the complete positivity of tensors more directly, such as to exclude the tensor in question from CP m,n , or to ensure the membership under certain algebraic operations that preserve the complete positivity for tensors.…”

“…In recent years, an emerging interest in the assets of multi-linear algebra has been concentrated on the higher-order tensors, which serve as a numerical tool, complementary to the arsenal of existing matrix techniques. In this vein, the concept of completely positive matrices has been extended to higher-order completely positive tensors, which are connected with nonnegative tensor factorization and have wide applications in statistics, computer vision, exploratory multiway data analysis, blind source separation and higher degree polynomial optimization [15,20,33,35]. As an extension of the completely positive matrix, a completely positive tensor admits its definition in a pretty natural way as initiated by Qi et al in [33] and recalled below.…”

“…Spectral properties, dominance properties, the Hadamard product preservation property, and even a special structured subclass were proposed for CP tensors [30,33], which can be served as necessary or sufficient conditions for CP tensor verification. An optimization algorithm based on semidefinite relaxation was even proposed by Fan and Zhou in their recent work [20], from which either a certificate would be provided for non-CP tensors or a numerical CP decomposition would be obtained. Additionally, numerical optimization for the best fit of CP tensors with given length of decomposition was formulated as a nonnegative constrained least-squares problem in Kolda's more recent paper [22].…”

Numerical optimization for symmetric tensor decomposition

“…Our formulation finds the best nonnegative factorization. Fan and Zhou [20] consider the problem of verifying that a tensor is completely positive.…”