Hankel tensors, Vandermonde tensors and their positivities

Xu, Changqing

doi:10.1016/j.laa.2015.02.012

Cited by 23 publications

(14 citation statements)

References 16 publications

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“…The concept of Hankel tensor was introduced by Qi [25]. For more details on Hankel tensors, see Qi [25], Chen and Qi [5], Xu [37]. Denote such an mth-order (n + 1)dimensional generalized Hilbert tensor by H n λ .…”

Section: Juan Meng and Yisheng Songmentioning

confidence: 99%

Upper bounds for Z<inline-formula><tex-math id="M1">\begin{document}$ _1 $\end{document}</tex-math></inline-formula>-eigenvalues of generalized Hilbert tensors

Meng¹,

Song²

2020

Journal of Industrial &Amp; Management Optimization

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The Z 1-eigenvalue of tensors (hypermatrices) was widely used to discuss the properties of higher order Markov chains and transition probability tensors. In this paper, we extend the concept of Z 1-eigenvalue from finite-dimensional tensors to infinite-dimensional tensors, and discuss the upper bound of such eigenvalues for infinite-dimensional generalized Hilbert tensors. Furthermore, an upper bound of Z 1-eigenvalue for finite-dimensional generalized Hilbert tensor is obtained also.

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Section: Juan Meng and Yisheng Songmentioning

confidence: 99%

Upper bounds for Z<inline-formula><tex-math id="M1">\begin{document}$ _1 $\end{document}</tex-math></inline-formula>-eigenvalues of generalized Hilbert tensors

Meng¹,

Song²

2020

Journal of Industrial &Amp; Management Optimization

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“…Example 3 A Vandermonde tensor [43,53] is a special Hankel tensor. Let α = n n − 1 and β = 1 − n n .…”

Section: Large Scale Problemsmentioning

confidence: 99%

“…On the mathematical properties, Luque and Thibon [34] explored the Hankel hyperdeterminants. Qi [43] and Xu [53] studied the spectra of Hankel tensors and gave some upper bounds and lower bounds for the smallest and the largest eigenvalues. In [43], Qi raised a question: Can we construct some efficient algorithms for the largest and the smallest H-and Z-eigenvalues of a Hankel tensor?…”

Section: Introductionmentioning

confidence: 99%

Computing Extreme Eigenvalues of Large Scale Hankel Tensors

Chen

Wang

2016

J Sci Comput

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Large scale tensors, including large scale Hankel tensors, have many applications in science and engineering. In this paper, we propose an inexact curvilinear search optimization method to compute Z-and H-eigenvalues of mth order n dimensional Hankel tensors, where n is large. Owing to the fast Fourier transform, the computational cost of each iteration of the new method is about O(mn log(mn)). Using the Cayley transform, we obtain an effective curvilinear search scheme. Then, we show that every limiting point of iterates generated by the new algorithm is an eigen-pair of Hankel tensors. Without the assumption of a second-order sufficient condition, we analyze the linear convergence rate of iterate sequence by the Kurdyka-Lojasiewicz property. Finally, numerical experiments for Hankel tensors, whose dimension may up to one million, are reported to show the efficiency of the proposed curvilinear search method.

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“…Clearly, an m-order n-dimensional Hilbert tensor is a Hankel tensor with v = (1, 1 2 , 1 3 , · · · , 1 nm ), introduced by Qi [19]. Also see Chen and Qi [3], Xu [26] for more details of Hankel tensors. Hilbert tensor (hypermatrix) is a natural extension of Hilbert matrix, which was introduced by Hilbert [7].…”

Section: Introductionmentioning

confidence: 99%

Infinite and finite dimensional generalized Hilbert tensors

Mei

Song

2017

Linear Algebra and its Applications

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In this paper, we introduce the concept of an $m$-order $n$-dimensional generalized Hilbert tensor $\mathcal{H}_{n}=(\mathcal{H}_{i_{1}i_{2}\cdots i_{m}})$, $$ \mathcal{H}_{i_{1}i_{2}\cdots i_{m}}=\frac{1}{i_{1}+i_{2}+\cdots i_{m}-m+a},\ a\in \mathbb{R}\setminus\mathbb{Z}^-;\ i_{1},i_{2},\cdots,i_{m}=1,2,\cdots,n, $$ and show that its $H$-spectral radius and its $Z$-spectral radius are smaller than or equal to $M(a)n^{m-1}$ and $M(a)n^{\frac{m}{2}}$, respectively, here $M(a)$ is a constant only dependent on $a$. Moreover, both infinite and finite dimensional generalized Hilbert tensors are positive definite for $a\geq1$. For an $m$-order infinite dimensional generalized Hilbert tensor $\mathcal{H}_{\infty}$ with $a>0$, we prove that $\mathcal{H}_{\infty}$ defines a bounded and positively $(m-1)$-homogeneous operator from $l^{1}$ into $l^{p}\ (1

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Hankel tensors, Vandermonde tensors and their positivities

Cited by 23 publications

References 16 publications

Upper bounds for Z<inline-formula><tex-math id="M1">\begin{document}$ _1 $\end{document}</tex-math></inline-formula>-eigenvalues of generalized Hilbert tensors

Upper bounds for Z<inline-formula><tex-math id="M1">\begin{document}$ _1 $\end{document}</tex-math></inline-formula>-eigenvalues of generalized Hilbert tensors

Computing Extreme Eigenvalues of Large Scale Hankel Tensors

Infinite and finite dimensional generalized Hilbert tensors

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