Author:  Chen, Haibin 
Title:  Structured tensors : theory and applications 
Degree:  Ph.D. 
Year:  2016 
Subject:  Imaging systems  Mathematics. Calculus of tensors Hong Kong Polytechnic University  Dissertations 
Department:  Dept. of Applied Mathematics 
Pages:  xx, 151 pages : color illustrations 
Language:  English 
InnoPac Record:  http://library.polyu.edu.hk/record=b2925558 
URI:  http://theses.lib.polyu.edu.hk/handle/200/8706 
Abstract:  The thesis is devoted to studying spectral properties and positive semidefiniteness of several kinds of structured tensors. Furthermore, the SOS (sumofsquares) tensor decomposition of structured tensors in the literature are established. Five topics are considered: 1. Positive definiteness and semidefiniteness of even order Cauchy tensors. 2. Generalized Cauchy tensors and Hankel tensors. 3. Some spectral properties of oddbipartite ZTensors and their absolute tensors. 4. SOS tensor decomposition and applications. 5. Positive semidefiniteness and extremal Heigenvalues of extended essentially nonnegative tensors. For topic 1, motivated by symmetric Cauchy matrices, we define symmetric Cauchy tensors and their generating vectors in this thesis. An even order symmetric Cauchy tensor is positive semidefinite if and only if its generating vector is positive. An even order symmetric Cauchy tensor is positive definite if and only if its generating vector has positive and mutually distinct entries. This extends Fiedler's result for symmetric Cauchy matrices to symmetric Cauchy tensors. Then, it is proven that the positive semidefiniteness character of an even order symmetric Cauchy tensor can be equivalently checked by the monotone increasing property of a homogeneous polynomial related to the Cauchy tensor. The homogeneous polynomial is strictly monotone increasing in the nonnegative orthant of the Euclidean space when the even order symmetric Cauchy tensor is positive definite. At last, bounds of the largest Heigenvalue of a positive semidefinite symmetric Cauchy tensor are given and several spectral properties on Zeigenvalues of odd order symmetric Cauchy tensors are shown. We also establish that all the Heigenvalues of nonnegative Cauchy tensors are nonnegative. Further questions on Cauchy tensors are raised. For topic 2, we present various new results on generalized Cauchy tensors and Hankel tensors. We first introduce the concept of generalized Cauchy tensors which extends Cauchy tensors in the current literature, and provide several conditions characterizing positive semidefiniteness of generalized Cauchy tensors with nonzero entries. Furthermore, we prove that all even order generalized Cauchy tensors with positive entries are completely positive tensors, which means every such that generalized Cauchy tensor can be decomposed as the sum of nonnegative rank1 tensors. Secondly, we present new mathematical properties of Hankel tensors. We prove that an even order Hankel tensor is Vandermonde positive semidefinite if and only if its associated plane tensor is positive semidefinite. We also show that, if the Vandermonde rank of a Hankel tensor A is less than the dimension of the underlying space, then positive semidefiniteness of A is equivalent to the fact that A is a complete Hankel tensor, and so, is further equivalent to the SOS tensor decomposition property of A. Thirdly, we introduce a new class of structured tensors called CauchyHankel tensors, which is a special case of Cauchy tensors and Hankel tensors simultaneously. Sufficient and necessary conditions are established for an even order CauchyHankel tensor to be positive definite. For topic 3, stimulated by oddbipartite and evenbipartite hypergraphs, we define oddbipartite (weakly oddbipartite) and evenbipartite (weakly evenbipartite) tensors. It is verified that all even order oddbipartite tensors are irreducible tensors, while all evenbipartite tensors are reducible no matter the parity of the order. Based on properties of oddbipartite tensors, we study the relationship between the largest Heigenvalue of a symmetric Ztensor with nonnegative diagonal elements, and the largest Heigenvalue of absolute tensor of that Ztensor. When the order is even and the symmetric Ztensor is weakly irreducible, we prove that the largest Heigenvalue of the Ztensor and the largest Heigenvalue of the absolute tensor of that Ztensor are equal, if and only if the Ztensor is weakly oddbipartite. Examples show the authenticity of the conclusions. Then, we prove that a symmetric Ztensor with nonnegative diagonal entries and the absolute tensor of the Ztensor are diagonal similar, if and only if the Ztensor has even order and it is weakly oddbipartite. After that, it is proved that, when an even order symmetric Ztensor with nonnegative diagonal entries is weakly irreducible, the equality of the spectrum of the Ztensor and the spectrum of absolute tensor of that Ztensor, can be characterized by the equality of their spectral radii. For topic 4, we examine structured tensors which have SOS tensor decomposition, and study the SOSrank of SOS tensor decomposition. We first show that several classes of even order symmetric structured tensors available in the literature have SOS tensor decomposition. These include positive Cauchy tensors, weakly diagonally dominated tensors, B0tensors, double Btensors, quasidouble B0tensors, MB0 tensors, Htensors, absolute tensors of positive semidefinite Ztensors and extended Ztensors. We also examine the SOSrank of SOS tensor decomposition and the SOSwidth for SOS tensor cones. The SOSrank provides the minimal number of squares in the SOS tensor decomposition, and, for a given SOS tensor cone, its SOSwidth is the maximum possible SOSrank for all the tensors in this cone. We first deduce an upper bound for general tensors that have SOS decomposition and the SOSwidth for general SOS tensor cone using the known results in the literature of polynomial theory. Then, we provide an explicit sharper estimate for the SOSrank of SOS tensor decomposition with bounded exponent and identify the SOSwidth for the tensor cone consisting of all tensors with bounded exponent that have SOS decompositions. Finally, as applications, we show how the SOS tensor decomposition can be used to compute the minimum Heigenvalue of an even order symmetric extended Ztensor and test the positive definiteness of an associated multivariate form. Numerical experiments are also provided to show the efficiency of the proposed numerical methods ranging from small size to large size numerical examples. For topic 5, we study positive semidefiniteness and extremal Heigenvalues of extended essentially nonnegative tensors. We first prove that checking positive semidefiniteness of a symmetric extended essentially nonnegative tensor is equivalent to checking positive semidefiniteness of all its condensed subtensors. Then, we prove that, for a symmetric positive semidefinite extended essentially nonnegative tensor, it has a sumofsquares (SOS) tensor decomposition if each positive offdiagonal element corresponds to an SOS term in the homogeneous polynomial of the tensor. Using this result, we can compute the minimum Heigenvalue of such kinds of extended essentially nonnegative tensors. Then, for general symmetric even order extended essentially nonnegative tensors, we show that the largest Heigenvalue of the tensor is equivalent to the optimal value of an SOS programming problem. As an application, we show this approach can be used to check copositivity of symmetric extended Ztensors. Numerical experiments are given to show the efficiency of the proposed methods. 
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