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**1 - 7**of**7**### HANKEL TENSORS: ASSOCIATED HANKEL MATRICES AND VANDERMONDE DECOMPOSITION∗

"... Abstract. 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 a Hankel matrix and a higher order two-dimensional symmetric tensor, which we call the associated plane tensor. If the associate ..."

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Abstract. 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 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-eigenvalues 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.

### Magnetic Resonance Imaging (MRI) [6], especially Diffusion Tensor Imaging

"... (Communicated by Hao-Min Zhou and Yunmei Chen) Abstract. Diffusion Kurtosis Imaging (DKI) is a new Magnetic Resonance Imaging (MRI) model to characterize the non-Gaussian diffusion behavior in tissues. In reality, the term bDapp − 1 6 b2D2appKapp in the extended Stejskal and Tanner equation of DKI s ..."

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(Communicated by Hao-Min Zhou and Yunmei Chen) Abstract. Diffusion Kurtosis Imaging (DKI) is a new Magnetic Resonance Imaging (MRI) model to characterize the non-Gaussian diffusion behavior in tissues. In reality, the term bDapp − 1 6 b2D2appKapp in the extended Stejskal and Tanner equation of DKI should be positive for an appropriate range of b-values to make sense physically. The positive definiteness of the above term reflects the signal attenuation in tissues during imaging. Hence, it is essential for the validation of DKI. In this paper, we analyze the positive definiteness of DKI. We first char-acterize the positive definiteness of DKI through the positive definiteness of a tensor constructed by diffusion tensor and diffusion kurtosis tensor. Then, a conic linear optimization method and its simplified version are proposed to handle the positive definiteness of DKI from the perspective of numerical com-putation. Some preliminary numerical tests on both synthetical and real data show that the method discussed in this paper is promising.

### A Tensor Analogy of Yuan’s Theorem of the Alternative and Polynomial Optimization with Sign Structure

"... Yuan’s theorem of the alternative is an important theoretical tool in optimization, which provides a checkable certificate for the infeasibility of a strict inequality system involving two homogeneous quadratic functions. In this paper, we provide a tractable extension of Yuan’s theorem of the alter ..."

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Yuan’s theorem of the alternative is an important theoretical tool in optimization, which provides a checkable certificate for the infeasibility of a strict inequality system involving two homogeneous quadratic functions. In this paper, we provide a tractable extension of Yuan’s theorem of the alternative to the symmetric tensor setting. As an application, we establish that the optimal value of a class of nonconvex polynomial optimization problems with suitable sign structure (or more explicitly, with essentially non-positive coefficients) can be computed by a related convex conic programming problem, and the optimal solution of these nonconvex polynomial optimization problems can be recovered from the corresponding solution of the convex conic programming problem. Moreover, we obtain that this class of nonconvex polynomial optimization problems enjoy exact sum-of-squares relaxation, and so, can be solved via a single semidefinite programming problem.

### Linear operators and positive semidefiniteness of symmetric tensor spaces

, 2013

"... Abstract We study symmetric tensor spaces and cones arising from polynomial optimization and physical sciences. We prove a decomposition invariance theorem for linear operators over the symmetric tensor space, which leads to several other interesting properties in symmetric tensor spaces. We then co ..."

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Abstract We study symmetric tensor spaces and cones arising from polynomial optimization and physical sciences. We prove a decomposition invariance theorem for linear operators over the symmetric tensor space, which leads to several other interesting properties in symmetric tensor spaces. We then consider the positive semidefiniteness of linear operators which deduces the convexity of the Frobenius norm function of a symmetric tensor. Furthermore, we characterize the symmetric positive semidefinite tensor (SDT) cone by employing the properties of linear operators, design some face structures of its dual cone, and analyze its relationship to many other tensor cones. In particular, we show that the cone is self-dual if and only if the polynomial is quadratic, give specific characterizations of tensors that are in the primal cone but not in the dual for higher order cases, and develop a complete relationship map among the tensor cones appeared in the literature. Keywords symmetric tensor, symmetric positive semidefinite tensor cone, linear operator, SOS cone