Minimax Theory and its Applications 05 (2020), No. 1, 033--046
Copyright Heldermann Verlag 2020
Basic Positive Semi-Definite Hankel Tensors
Lejia Gu
Dept. of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong
le-jia.gu@connect.polyu.hk
Liqun Qi
Dept. of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong
liqun.qi@polyu.edu.hk
Some classes of positive semi-definite Hankel tensors which are not strong Hankel tensors were recently introduced by Q. Wang, G. Li, L. Qi and Y. Xu [New classes of positive semi-definite Hankel tensors, Minimax Theory and its Applications 2 (2017) 231-248]. In this paper, we continue the study of such tensors.
We introduce a subclass of Hankel tensors called basic positive semi-definite Hankel tensors and intend to find some low-rank basic PSD non-strong Hankel tensors. We show that rank-1 even order strong Hankel tensors are equivalent to rank-1 basic PSD Hankel tensors, and all even order strong Hankel tensors with rank larger than 1 can be expressed as the sum of rank-1 basic PSD Hankel tensors. Thus, the study of non-strong PSD Hankel tensors is reduced to the study of basic PSD Hankel tensors with rank larger than 1. We prove that (1) there are no rank-2 basic PSD Hankel tensors, (2) rank-3 basic PSD Hankel tensors with dimension no less than 3 do not exist. Furthermore, an example of low-rank basic PSD Hankel tensor whose rank equals 3 or 4 is provided.
Keywords: Hankel tensors, basic positive semi-definite Hankel tensors, symmetric rank, Vandermonde decomposition.
MSC: 15A03, 15A69, 53A45.
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