Structured tensors : theory and applications

Abstract

The thesis is devoted to studying spectral properties and positive semi-definiteness of several kinds of structured tensors. Furthermore, the SOS (sum-of-squares) tensor decomposition of structured tensors in the literature are established. Five topics are considered: 1. Positive definiteness and semi-definiteness of even order Cauchy tensors. 2. Generalized Cauchy tensors and Hankel tensors. 3. Some spectral properties of odd-bipartite Z-Tensors and their absolute tensors. 4. SOS tensor decomposition and applications. 5. Positive semi-definiteness and extremal H-eigenvalues of extended essentially non-negative 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 semi-definite 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 semi-definiteness 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 non-negative orthant of the Euclidean space when the even order symmetric Cauchy tensor is positive definite. At last, bounds of the largest H-eigenvalue of a positive semi-definite symmetric Cauchy tensor are given and several spectral properties on Z-eigenvalues of odd order symmetric Cauchy tensors are shown. We also establish that all the H-eigenvalues of non-negative Cauchy tensors are non-negative. 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 semi-definiteness 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 non-negative rank-1 tensors. Secondly, we present new mathematical properties of Hankel tensors. We prove that an even order Hankel tensor is Vandermonde positive semi-definite if and only if its associated plane tensor is positive semi-definite. We also show that, if the Vandermonde rank of a Hankel tensor A is less than the dimension of the underlying space, then positive semi-definiteness 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 Cauchy-Hankel tensors, which is a special case of Cauchy tensors and Hankel tensors simultaneously. Sufficient and necessary conditions are established for an even order Cauchy-Hankel tensor to be positive definite. For topic 3, stimulated by odd-bipartite and even-bipartite hypergraphs, we define odd-bipartite (weakly odd-bipartite) and even-bipartite (weakly even-bipartite) tensors. It is verified that all even order odd-bipartite tensors are irreducible tensors, while all even-bipartite tensors are reducible no matter the parity of the order. Based on properties of odd-bipartite tensors, we study the relationship between the largest H-eigenvalue of a symmetric Z-tensor with non-negative diagonal elements, and the largest H-eigenvalue of absolute tensor of that Z-tensor. When the order is even and the symmetric Z-tensor is weakly irreducible, we prove that the largest H-eigenvalue of the Z-tensor and the largest H-eigenvalue of the absolute tensor of that Z-tensor are equal, if and only if the Z-tensor is weakly odd-bipartite. Examples show the authenticity of the conclusions. Then, we prove that a symmetric Z-tensor with non-negative diagonal entries and the absolute tensor of the Z-tensor are diagonal similar, if and only if the Z-tensor has even order and it is weakly odd-bipartite. After that, it is proved that, when an even order symmetric Z-tensor with non-negative diagonal entries is weakly irreducible, the equality of the spectrum of the Z-tensor and the spectrum of absolute tensor of that Z-tensor, 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 SOS-rank 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, B0-tensors, double B-tensors, quasi-double B0-tensors, MB0 -tensors, H-tensors, absolute tensors of positive semi-definite Z-tensors and extended Z-tensors. We also examine the SOS-rank of SOS tensor decomposition and the SOS-width for SOS tensor cones. The SOS-rank provides the minimal number of squares in the SOS tensor decomposition, and, for a given SOS tensor cone, its SOS-width is the maximum possible SOS-rank for all the tensors in this cone. We first deduce an upper bound for general tensors that have SOS decomposition and the SOS-width for general SOS tensor cone using the known results in the literature of polynomial theory. Then, we provide an explicit sharper estimate for the SOS-rank of SOS tensor decomposition with bounded exponent and identify the SOS-width 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 H-eigenvalue of an even order symmetric extended Z-tensor 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 semi-definiteness and extremal H-eigenvalues of extended essentially non-negative tensors. We first prove that checking positive semi-definiteness of a symmetric extended essentially non-negative tensor is equivalent to checking positive semi-definiteness of all its condensed subtensors. Then, we prove that, for a symmetric positive semi-definite extended essentially non-negative tensor, it has a sum-of-squares (SOS) tensor decomposition if each positive off-diagonal element corresponds to an SOS term in the homogeneous polynomial of the ten-sor. Using this result, we can compute the minimum H-eigenvalue of such kinds of extended essentially non-negative tensors. Then, for general symmetric even order extended essentially non-negative tensors, we show that the largest H-eigenvalue 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 co-positivity of symmetric extended Z-tensors. Numerical experiments are given to show the efficiency of the proposed methods.