Further results on Cauchy tensors and Hankel tensors

Chen, Haibin; Li, Guoyin; Qi, Liqun

doi:10.1016/j.amc.2015.11.051

Cited by 11 publications

(4 citation statements)

References 42 publications

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“…Extremal eigenvalues of Hankel tensors are discussed in Chen, Qi and Wang [8]. Some further results about Hankel tensors appear in Chen, Li and Qi [7].…”

Section: Hankel Tensors a Tensormentioning

confidence: 97%

Hankel Tensor Decompositions and Ranks

Nie¹,

Ye²

2019

SIAM J. Matrix Anal. Appl.

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Hermitian tensors are generalizations of Hermitian matrices, but they have very different properties. Every complex Hermitian tensor is a sum of complex Hermitian rank-1 tensors. However, this is not true for the real case. We study basic properties for Hermitian tensors such as Hermitian decompositions and Hermitian ranks. For canonical basis tensors, we determine their Hermitian ranks and decompositions. For real Hermitian tensors, we give a full characterization for them to have Hermitian decompositions over the real field. In addition to traditional flattening, Hermitian tensors specially have Hermitian and Kronecker flattenings, which may give different lower bounds for Hermitian ranks. We also study other topics such as eigenvalues, positive semidefiniteness, sum of squares representations, and separability.2010 Mathematics Subject Classification. 15A69,15B48,65F99.

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“…Extremal eigenvalues of Hankel tensors are discussed in Chen, Qi and Wang [8]. Some further results about Hankel tensors appear in Chen, Li and Qi [7].…”

Section: Hankel Tensors a Tensormentioning

confidence: 97%

Hankel Tensor Decompositions and Ranks

Nie¹,

Ye²

2019

SIAM J. Matrix Anal. Appl.

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“…The implication "(ii) ⇒ (iii)" follows readily from 0 < Ce m i = 1 mci for any i ∈ [n]. To get "(iii) ⇒ (i)", we can employ the proof in Theorem 3.1 in [11] that for any x ∈ R n , it yields that…”

Section: Positive Cauchy Tensorsmentioning

confidence: 99%

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

Luo¹,

Qi²

2016

SIAM J. Matrix Anal. & Appl.

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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.

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“…Ding, Qi and Wei [14] studied inheritance properties of Hankel tensors. The relations to Cauchy and Vandermonde tensors are discussed in [8,33,34]. The ranks and decompositions of Hankel tensors are discussed in [23].…”

Section: Introductionmentioning

confidence: 99%

Completely Positive Binary Tensors

2019

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A symmetric tensor is completely positive (CP) if it is a sum of tensor powers of nonnegative vectors. This paper characterizes completely positive binary tensors. We show that a binary tensor is completely positive if and only if it satisfies two linear matrix inequalities. This result can be used to determine whether a binary tensor is completely positive or not. When it is, we give an algorithm for computing its cp-rank and the decomposition. When the order is odd, we show that the cp-rank decomposition is unique. When the order is even, we completely characterize when the cp-rank decomposition is unique. We also discuss how to compute the nearest cp-approximation when a binary tensor is not completely positive.2010 Mathematics Subject Classification. Primary 15A69, 44A60, 90C22.

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Further results on Cauchy tensors and Hankel tensors

Cited by 11 publications

References 42 publications

Hankel Tensor Decompositions and Ranks

Hankel Tensor Decompositions and Ranks

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

Completely Positive Binary Tensors

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