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.
“…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.…”
Section: Definition 22 Suppose That a ∈ T Nimentioning
The main purpose of this paper is to investigate inequalities on symmetric sums of diagonalizable and positive definite tensors. In particular, we generalize the well-known Hlawka and Popoviciu inequalities to the case of diagonalizable and positive definite tensors. As corollaries, we extend Hlawka and Popoviciu inequalities for the combinatorial determinant, permanent and immanant of tensors, and generalized tensor functions.
“…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.…”
Section: Definition 22 Suppose That a ∈ T Nimentioning
The main purpose of this paper is to investigate inequalities on symmetric sums of diagonalizable and positive definite tensors. In particular, we generalize the well-known Hlawka and Popoviciu inequalities to the case of diagonalizable and positive definite tensors. As corollaries, we extend Hlawka and Popoviciu inequalities for the combinatorial determinant, permanent and immanant of tensors, and generalized tensor functions.
“…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.…”
Section: Definition 22 ([29])mentioning
confidence: 99%
“…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.…”
Section: Introductionmentioning
confidence: 99%
“…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].…”
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.
“…Our formulation finds the best nonnegative factorization. Fan and Zhou [20] consider the problem of verifying that a tensor is completely positive.…”
Section: Optimization Formulation For Nonnegative Symmetric Factorizamentioning
We consider the problem of decomposing a real-valued symmetric tensor as the sum of outer products of real-valued vectors. Algebraic methods exist for computing complex-valued decompositions of symmetric tensors, but here we focus on real-valued decompositions, both unconstrained and nonnegative, for problems with low-rank structure. We discuss when solutions exist and how to formulate the mathematical program. Numerical results show the properties of the proposed formulations (including one that ignores symmetry) on a set of test problems and illustrate that these straightforward formulations can be effective even though the problem is nonconvex.
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