Hankel Tensors: Associated Hankel Matrices and Vandermonde Decomposition

By Liqun Qi. 2013

85 citations recorded by the source corpus.

Hankel tensors arise from applications such as signal processing. In this paper, we make an initial study on Hankel tensors. For each Hankel tensor, we associate it with a Hankel matrix and a higher order two-dimensional symmetric tensor, which we call the associated plane tensor. If the associated Hankel matrix is positive semi-definite, we call such a Hankel tensor a strong Hankel tensor. We show that an $m$ order $n$-dimensional tensor is a Hankel tensor if and only if it has a Vandermonde decomposition. We call a Hankel tensor a complete Hankel tensor if it has a Vandermonde decomposition with positive coefficients. We prove that if a Hankel tensor is copositive or an even order Hankel tensor is positive semi-definite, then the associated plane tensor is copositive or positive semi-definite, respectively. We show that even order strong and complete Hankel tensors are positive semi-definite, the Hadamard product of two strong Hankel tensors is a strong Hankel tensor, and the Hadamard product of two complete Hankel tensors is a complete Hankel tensor. We show that all the H-eigenvalue of a complete Hankel tensors (maybe of odd order) are nonnegative. We give some upper bounds and lower bounds for the smallest and the largest Z-eigenvalues of a Hankel tensor, respectively. Further questions on Hankel tensors are raised.

Subfields: Computational Mathematics, Computational Theory and Mathematics, Signal Processing

View Paper: https://www.semanticscholar.org/paper/5c9cca5ece8b0b482b4a8e3e3289b511a6aadd4e

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In Computational Mathematics

1. Chaoqian Li
2. Yaotang Li
3. Yisheng Song
4. Haibin Chen
5. Zhongming Chen
6. Liqun Qi
7. J. Shao
8. Guanglu Zhou

In Computational Theory and Mathematics

1. Chaoqian Li
2. Yaotang Li
3. Yisheng Song
4. Haibin Chen
5. Zhongming Chen
6. Liqun Qi
7. Guanglu Zhou
8. J. Shao

In Signal Processing

1. Yisheng Song
2. Chaoqian Li
3. Yaotang Li
4. Haibin Chen
5. Zhongming Chen
6. Liqun Qi

Citations in the corpus

• An eigenvalue localization set for tensors with applications to determine the positive (semi-)definiteness of tensors (2016, background): “[1,4,5] Eigenvalue problems of tensors have become an important topic of study in numerical multilinear algebra, and they have a wide range of practical applications; see, [5–30].”
• An even order symmetric B tensor is positive definite (2014, background): “This includes diagonally dominated tensors discussed in Section 2 of this paper, complete Hankel tensors and strong Hankel tensors [19], completely positive tensors [21] and P0 tensors [26].”
• Infinite and finite dimensional Hilbert tensors (2014, background): “Clearly, both Hn and H∞ are positive (Hi1i2···im > 0) and symmetric (Hi1i2···im are invariant for any permutation of the indices), and an m-order n-dimensional Hilbert tensor Hn is a Hankel tensor with v = (1, 12 , 13 , · · · , 1 nm ) (Qi [13]), and an m-order infinite dimensional Hilbert tensor H∞ ”
• Positive Definiteness and Semi-Definiteness of Even Order Symmetric Cauchy Tensors (2014, background): “In recent years, a lot of research papers on structured tensors are presented [2, 3, 4, 8, 13, 14, 15, 17, 18, 21, 22], which studied M-tensors, circulant tensors, completely positive tensors, Hankel tensors, Hilbert tensors, P tensors and B tensors.”
• Tensor Networks for Dimensionality Reduction and Large-scale Optimization: Part 2 Applications and Future Perspectives (2017, unknown): “Moreover, the tensor Y also admits a symmetric CP decomposition with Vandermonde structured factor matrix [Qi, 2015] Y = diagN(λ)ˆ1 V T ˆ2 VT ¨ ¨ ¨ ˆN VT , (1.19) where λ comprises R non-zero coefficients, and V is a Vandermonde matrix generated from R distinct values v = [v1, v2, . . . , vR] V = …”
• Circulant Tensors with Applications to Spectral Hypergraph Theory and Stochastic Process (2013, background): “roves the theorem. ✷ In matrix theory [7], the Shur Product Theorem says that the Hadamard product of two positive semidefinite symmetric matrices is still a positive semi-definite symmetric matrix. In [12], Hankel tensors are studied. Hankel tensors are symmetric tensors. The Hadamard product of two ”