Completely positive tensor decomposition

Fan, Junxuan; Zhou, Anwa

doi:10.48550/arxiv.1411.5149

Cited by 4 publications

(5 citation statements)

References 23 publications

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“…As said before, Fan and Zhou [21] proposed an optimization algorithm to decompose a completely positive tensor. This is an important contribution to the study of completely positive tensor decomposition.…”

Section: Cones Of Doubly Nonnegative Tensorsmentioning

confidence: 99%

“…The completely positive tensor verification and decomposition are very important as discussed in [21,28]. In this section, by employing the zero-entry dominance property and a simplified strong dominance property called the one-duplicated dominance property, a preprocessing scheme is proposed to accelerate the verification procedure for completely positive tensors based on the Fan-Zhou algorithm.…”

Section: Application 1: Preprocessing For Cp Tensorsmentioning

confidence: 99%

“…Analogous to the matrix case, completely positive tensors form an extremely important part of doubly nonnegative tensors. They are connected with nonnegative tensor factorization and have wide applications [12,21,43,46]. 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 [43] and recalled below.…”

Section: Introductionmentioning

confidence: 99%

“…Numerical optimization for the best fit of completely positive tensors with given length of decomposition was formulated as a nonnegative constrained least-squares problem in Kolda's paper [28]. A verification 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 their recent work [21]. In this paper, the properties, together with their relation and difference will also be investigated for the doubly nonnegative tensor cone and the completely positive tensor cone, which somehow shows a different phenomenon from the matrix case.…”

Section: Introductionmentioning

confidence: 99%

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Doubly Nonnegative Tensors, Completely Positive Tensors and Applications

Luo,

2015

Preprint

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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.

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Section: Cones Of Doubly Nonnegative Tensorsmentioning

confidence: 99%

Section: Application 1: Preprocessing For Cp Tensorsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Doubly Nonnegative Tensors, Completely Positive Tensors and Applications

Luo,

2015

Preprint

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“…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 Factoriza...mentioning

confidence: 99%

Numerical optimization for symmetric tensor decomposition

Kolda

2015

Math. Program.

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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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