Inheritance properties and sum-of-squares decomposition of Hankel tensors: theory and algorithms

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• 作者：Weiyang Ding ; Liqun Qi ; Yimin Wei
• 关键词：Hankel tensor ; Inheritance property ; Positive semi ; definite tensor ; Sum ; of ; squares ; Convolution
• 刊名：BIT Numerical Mathematics
• 出版年：2017
• 出版时间：March 2017
• 年：2017
• 卷：57
• 期：1
• 页码：169-190
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Computational Mathematics and Numerical Analysis; Numeric Computing; Mathematics, general;
• 出版者：Springer Netherlands
• ISSN：1572-9125
• 卷排序：57

文摘

In this paper, we show that if a lower-order Hankel tensor is positive semi-definite (or positive definite, or negative semi-definite, or negative definite, or SOS), then its associated higher-order Hankel tensor with the same generating vector, where the higher order is a multiple of the lower order, is also positive semi-definite (or positive definite, or negative semi-definite, or negative definite, or SOS, respectively). Furthermore, in this case, the extremal H-eigenvalues of the higher order tensor are bounded by the extremal H-eigenvalues of the lower order tensor, multiplied with some constants. Based on this inheritance property, we give a concrete sum-of-squares decomposition for each strong Hankel tensor. Then we prove the second inheritance property of Hankel tensors, i.e., a Hankel tensor has no negative (or non-positive, or positive, or nonnegative) H-eigenvalues if the associated Hankel matrix of that Hankel tensor has no negative (or non-positive, or positive, or nonnegative, respectively) eigenvalues. In this case, the extremal H-eigenvalues of the Hankel tensor are also bounded by the extremal eigenvalues of the associated Hankel matrix, multiplied with some constants. The third inheritance property of Hankel tensors is raised as a conjecture.