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Haibin Chen  One of the best experts on this subject based on the ideXlab platform.

Column sufficient Tensors and Tensor complementarity problems
Frontiers of Mathematics in China, 2018CoAuthors: Haibin Chen, Yisheng SongAbstract:Stimulated by the study of sufficient matrices in linear complementarity problems, we study column sufficient Tensors and Tensor complementarity problems. Column sufficient Tensors constitute a wide range of Tensors that include positive semidefinite Tensors as special cases. The inheritance property and invariant property of column sufficient Tensors are presented. Then, various spectral properties of symmetric column sufficient Tensors are given. It is proved that all Heigenvalues of an evenorder symmetric column sufficient Tensor are nonnegative, and all its Zeigenvalues are nonnegative even in the odd order case. After that, a new subclass of column sufficient Tensors and the handicap of Tensors are defined. We prove that a Tensor belongs to the subclass if and only if its handicap is a finite number. Moreover, several optimization models that are equivalent with the handicap of Tensors are presented. Finally, as an application of column sufficient Tensors, several results on Tensor complementarity problems are established.

Spectral properties of oddbipartite Z Tensors and their absolute Tensors
Frontiers of Mathematics in China, 2016CoAuthors: Haibin ChenAbstract:Stimulated by oddbipartite and evenbipartite hypergraphs, we define oddbipartite (weakly oddbipartie) 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 ZTensor with nonnegative diagonal elements, and the largest Heigenvalue of absolute Tensor of that ZTensor. When the order is even and the 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.

Further results on Cauchy Tensors and Hankel Tensors
Applied Mathematics and Computation, 2016CoAuthors: Haibin ChenAbstract:In this article, we present various new results on 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. We also establish that all the Heigenvalues of nonnegative Cauchy Tensors are nonnegative. 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 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.

Some Spectral Properties of OddBipartite $Z$Tensors and Their Absolute Tensors
arXiv: Spectral Theory, 2015CoAuthors: Haibin ChenAbstract:Stimulated by oddbipartite and evenbipartite hypergraphs, we define oddbipartite (weakly oddbipartie) 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 $Z$Tensor with nonnegative diagonal elements, and the largest Heigenvalue of absolute Tensor of that $Z$Tensor. When the order is even and the $Z$Tensor is weakly irreducible, we prove that the largest Heigenvalue of the $Z$Tensor and the largest Heigenvalue of the absolute Tensor of that $Z$Tensor are equal, if and only if the $Z$Tensor is weakly oddbipartite. Examples show the authenticity of the conclusions. Then, we prove that a symmetric $Z$Tensor with nonnegative 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 oddbipartite. After that, it is proved that, when an even order symmetric $Z$Tensor with nonnegative 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.

Further Results on Cauchy Tensors and Hankel Tensors
arXiv: Spectral Theory, 2015CoAuthors: Haibin ChenAbstract:In this article, we present various new results on 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.} As a consequence, we show that Cauchy Tensors are positive semidefinite if and only if they are SOS (Sumofsquares) Tensors.} Furthermore, we prove that all positive semidefinite Cauchy Tensors are completely positive Tensors, which means every positive semidefinite Cauchy Tensor can be decomposed { as} the sum of nonnegative rank1 Tensors. We also establish that all the Heigenvalues of nonnegative Cauchy Tensors are nonnegative. 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 $\mathcal{A}$ is less than the dimension of the underlying space, then positive semidefiniteness of $\mathcal{A}$ is equivalent to the fact that $\mathcal{A}$ is a complete Hankel Tensor, and so, is further equivalent to the SOS property of $\mathcal{A}$. Lastly, we introduce a new structured Tensor 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. Final remarks are listed at the end of the paper.
Yimin Wei  One of the best experts on this subject based on the ideXlab platform.

generalized Tensor function via the Tensor singular value decomposition based on the t product
Linear Algebra and its Applications, 2020CoAuthors: Yun Miao, Yimin WeiAbstract:Abstract In this paper, we present the definition of generalized Tensor function according to the Tensor singular value decomposition (TSVD) based on the Tensor Tproduct. Also, we introduce the compact singular value decomposition (TCSVD) of Tensors, from which the projection operators and MoorePenrose inverse of Tensors are obtained. We establish the Cauchy integral formula for Tensors by using the partial isometry Tensors and apply it into the solution of Tensor equations. Then we establish the generalized Tensor power and the Taylor expansion of Tensors. Explicit generalized Tensor functions are listed. We define the Tensor bilinear and sesquilinear forms and propose theorems on structures preserved by generalized Tensor functions. For complex Tensors, we established an isomorphism between complex Tensors and real Tensors. In the last part of our paper, we find that the block circulant operator establishes an isomorphism between Tensors and matrices. This isomorphism is used to prove the Fstochastic structure is invariant under generalized Tensor functions. The concept of invariant Tensor cones is raised.

Stochastic structured Tensors to stochastic complementarity problems
Computational Optimization and Applications, 2019CoAuthors: Maolin Che, Yimin WeiAbstract:This paper is concerned with the stochastic structured Tensors to stochastic complementarity problems. The definitions and properties of stochastic structured Tensors, such as the stochastic strong PTensors, stochastic PTensors, stochastic \(P_{0}\)Tensors, stochastic strictly semipositive Tensors and stochastic STensors are given. It is shown that the expected residual minimization formulation (ERM) of the stochastic structured Tensor complementarity problem has a nonempty and bounded solution set. Interestingly, we partially answer the open questions proposed by Che et al. (Optim Lett 13:261–279, 2019). We also consider the expected value method of stochastic structured Tensor complementarity problem with finitely many elements probability space. Finally, based on the expected residual minimization formulation (ERM) of the stochastic structured Tensor complementarity problem, a projected gradient method is proposed for solving the stochastic structured Tensor complementarity problem and the related numerical results are also given to show the efficiency of the proposed method.

generalized Tensor function via the Tensor singular value decomposition based on the t product
arXiv: Numerical Analysis, 2019CoAuthors: Yun Miao, Yimin WeiAbstract:In this paper, we present the definition of generalized Tensor function according to the Tensor singular value decomposition (TSVD) via the Tensor Tproduct. Also, we introduce the compact singular value decomposition (TCSVD) of Tensors via the Tproduct, from which the projection operators and Moore Penrose inverse of Tensors are also obtained. We also establish the Cauchy integral formula for Tensors by using the partial isometry Tensors and applied it into the solution of Tensor equations. Then we establish the generalized Tensor power and the Taylor expansion of Tensors. Explicit generalized Tensor functions are also listed. We define the Tensor bilinear and sesquilinear forms and proposed theorems on structures preserved by generalized Tensor functions. For complex Tensors, we established an isomorphism between complex Tensors and real Tensors. In the last part of our paper, we find that the block circulant operator established an isomorphism between Tensors and matrices. This isomorphism is used to prove the Fstochastic structure is invariant under generalized Tensor functions. The concept of invariant Tensor cones is also raised.

ℋTensors and nonsingular ℋTensors
Frontiers of Mathematics in China, 2015CoAuthors: Xuezhong Wang, Yimin WeiAbstract:The Hmatrices are an important class in the matrix theory, and have many applications. Recently, this concept has been extended to higher order ℋTensors. In this paper, we establish important properties of diagonally dominant Tensors and ℋTensors. Distributions of eigenvalues of nonsingular symmetric ℋTensors are given. An ℋ+Tensor is semipositive, which enlarges the area of semipositive Tensor from MTensor to ℋ+Tensor. The spectral radius of Jacobi Tensor of a nonsingular (resp. singular) ℋTensor is less than (resp. equal to) one. In particular, we show that a quasidiagonally dominant Tensor is a nonsingular ℋTensor if and only if all of its principal subTensors are nonsingular ℋTensors. An irreducible Tensor A is an ℋTensor if and only if it is quasidiagonally dominant.

Some results on the generalized inverse of Tensors and idempotent Tensors
arXiv: Numerical Analysis, 2014CoAuthors: Lizhu Sun, Baodong Zheng, Yimin WeiAbstract:Let $\mathcal{A}$ be an order $t$ dimension $m\times n\times \cdots \times n$ Tensor over complex field. In this paper, we study some {generalized inverses} of $\mathcal{A}$, the {$k$Tidempotent Tensors} and the idempotent Tensors based on the general Tensor product. Using the Tensor generalized inverse, some solutions of the equation $\mathcal{A}\cdot x^{t1}=b$ are given, where $x$ and $b$ are dimension $n$ and $m$ vectors, respectively. The {generalized inverses} of some block Tensors, the eigenvalues of {$k$Tidempotent Tensors} and idempotent Tensors are given. And the relation between the generalized inverses of Tensors and the $k$Tidempotent Tensors is also showed.
Zhenghai Huang  One of the best experts on this subject based on the ideXlab platform.

Positive definiteness of paired symmetric Tensors and elasticity Tensors
Journal of Computational and Applied Mathematics, 2018CoAuthors: Zhenghai HuangAbstract:Abstract In this paper, we consider higher order paired symmetric Tensors and strongly paired symmetric Tensors. Elasticity Tensors and higher order elasticity Tensors in solid mechanics are strongly paired symmetric Tensors. A (strongly) paired symmetric Tensor is said to be positive definite if the homogeneous polynomial defined by it is positive definite. Positive definiteness of elasticity and higher order elasticity Tensors is strong ellipticity in solid mechanics, which plays an important role in nonlinear elasticity theory. We mainly investigate positive definiteness of fourth order three dimensional and sixth order three dimensional (strongly) paired symmetric Tensors. We first show that the concerned (strongly) paired symmetric Tensor is positive definite if and only if its smallest M eigenvalue is positive. Second, we propose several necessary and sufficient conditions under which the concerned (strongly) paired symmetric Tensor is positive definite. Third, we study the conditions under which the homogeneous polynomial defined by a fourth order three dimensional or sixth order three dimensional (strongly) paired symmetric Tensor can be written as a sum of squares of polynomials, and further, propose several necessary and/or sufficient conditions to judge whether the concerned (strongly) paired symmetric Tensors are positive definite or not. Fourth, by using semidefinite relaxation we propose a sequential semidefinite programming method to compute the smallest M eigenvalue of a fourth order three dimensional (strongly) paired symmetric Tensor, by which we can check positive definiteness of the concerned Tensor. The preliminary numerical results confirm our theoretical findings.

Positive Definiteness of Paired Symmetric Tensors and Elasticity Tensors
arXiv: Rings and Algebras, 2017CoAuthors: Zhenghai HuangAbstract:In this paper, we consider higher order paired symmetric Tensors and strongly paired symmetric Tensors. Elasticity Tensors and higher order elasticity Tensors in solid mechanics are strongly paired symmetric Tensors. A (strongly) paired symmetric Tensor is said to be positive definite if the homogeneous polynomial defined by it is positive definite. Positive definiteness of elasticity and higher order elasticity Tensors is strong ellipticity in solid mechanics, which plays an important role in nonlinear elasticity theory. We mainly investigate positive definiteness of fourth order three dimensional and sixth order three dimensional (strongly) paired symmetric Tensors. After discussing some basic properties of these Tensors, we first show that the concerned (strongly) paired symmetric Tensor is positive definite if and only if its smallest $M$eigenvalue is positive. Second, we propose several necessary and sufficient conditions under which the concerned (strongly) paired symmetric Tensor is positive definite. Third, we study the conditions under which the homogeneous polynomial defined by a fourth order three dimensional or sixth order three dimensional (strongly) paired symmetric Tensor can be written as a sum of squares of polynomials, and further, propose several necessary and sufficient conditions to judge whether the concerned (strongly) paired symmetric Tensors are positive definite or not. Fourth, by using semidefinite relaxation we propose a method to compute the smallest $M$eigenvalue of a fourth order three dimensional (strongly) paired symmetric Tensor, by which we can check positive definiteness of the concerned Tensor.

Positive Definiteness of Paired Symmetric Tensors and Elasticity Tensors
arXiv: Rings and Algebras, 2017CoAuthors: Zhenghai HuangAbstract:In this paper, we consider higher order paired symmetric Tensors and strongly paired symmetric Tensors. Elasticity Tensors and higher order elasticity Tensors in solid mechanics are strongly paired symmetric Tensors. A (strongly) paired symmetric Tensor is said to be positive definite if the homogeneous polynomial defined by it is positive definite. Positive definiteness of elasticity and higher order elasticity Tensors is strong ellipticity in solid mechanics, which plays an important role in nonlinear elasticity theory. We mainly investigate positive definiteness of fourth order three dimensional and sixth order three dimensional (strongly) paired symmetric Tensors. We first show that the concerned (strongly) paired symmetric Tensor is positive definite if and only if its smallest $M$eigenvalue is positive. Second, we propose several necessary and sufficient conditions under which the concerned (strongly) paired symmetric Tensor is positive definite. Third, we study the conditions under which the homogeneous polynomial defined by a fourth order three dimensional or sixth order three dimensional (strongly) paired symmetric Tensor can be written as a sum of squares of polynomials, and further, propose several necessary and/or sufficient conditions to judge whether the concerned (strongly) paired symmetric Tensors are positive definite or not. Fourth, by using semidefinite relaxation we propose a sequential semidefinite programming method to compute the smallest $M$eigenvalue of a fourth order three dimensional (strongly) paired symmetric Tensor, by which we can check positive definiteness of the concerned Tensor. The preliminary numerical results demonstrate that our method is effective.

Exceptionally regular Tensors and Tensor complementarity problems
Optimization Methods and Software, 2016CoAuthors: Yong Wang, Zhenghai Huang, Xueli BaiAbstract:Recently, many structured Tensors are defined and their properties are discussed in the literature. In this paper, we introduce a new class of structured Tensors, called exceptionally regular ER Tensor, which is relevant to the Tensor complementarity problem TCP. We show that this class of Tensors is a wide class of Tensors which includes many important structured Tensors as its special cases. By constructing two examples, we demonstrate that an ERTensor can be, but not always, an RTensor. We also show that within the class of the semipositive Tensors, the class of ERTensors coincides with the class of RTensors. In particular, we consider the TCP with an ERTensor and show that its solution set is nonempty and compact. In addition, we also obtain that the solution sets of the TCP with an RTensor or a Tensor are nonempty and compact.

On $Q$Tensors
arXiv: Optimization and Control, 2015CoAuthors: Zhenghai Huang, Yunyang Suo, Jie WangAbstract:One of the central problems in the theory of linear complementarity problems (LCPs) is to study the class of $Q$matrices since it characterizes the solvability of LCP. Recently, the concept of $Q$matrix has been extended to the case of Tensor, called $Q$Tensor, which characterizes the solvability of the corresponding Tensor complementarity problem  a generalization of LCP; and some basic results related to $Q$Tensors have been obtained in the literature. In this paper, we extend two famous results related to $Q$matrices to the Tensor space, i.e., we show that within the class of strong $P_0$Tensors or nonnegative Tensors, four classes of Tensors, i.e., $R_0$Tensors, $R$Tensors, $ER$Tensors and $Q$Tensors, are all equivalent. We also construct several examples to show that three famous results related to $Q$matrices cannot be extended to the Tensor space; and one of which gives a negative answer to a question raised recently by Song and Qi.
Liqun Qi  One of the best experts on this subject based on the ideXlab platform.

elasticity m Tensors and the strong ellipticity condition
Applied Mathematics and Computation, 2020CoAuthors: Weiyang Ding, Liqun QiAbstract:Abstract In this paper, we establish two sufficient conditions for the strong ellipticity of any fourthorder elasticity Tensor and investigate a class of Tensors satisfying the strong ellipticity condition, the elasticity M Tensor. The first sufficient condition is that the strong ellipticity holds if the unfolding matrix of this fourthorder elasticity Tensor can be modified into a positive definite one by preserving the summations of some corresponding entries. Second, an alternating projection algorithm is proposed to verify whether an elasticity Tensor satisfies the first condition or not. Besides, the elasticity M Tensor is defined with respect to the Meigenvalues of elasticity Tensors. We prove that any nonsingular elasticity M Tensor satisfies the strong ellipticity condition by employing a PerronFrobeniustype theorem for Mspectral radii of nonnegative elasticity Tensors. Other equivalent definitions of nonsingular elasticity M Tensors are also established.

elasticity mathscr m Tensors and the strong ellipticity condition
arXiv: Mathematical Physics, 2017CoAuthors: Weiyang Ding, Liqun QiAbstract:In this paper, we establish two sufficient conditions for the strong ellipticity of any fourthorder elasticity Tensor and investigate a class of Tensors satisfying the strong ellipticity condition, the elasticity $\mathscr{M}$Tensor. The first sufficient condition is that the strong ellipticity holds if the unfolding matrix of this fourthorder elasticity Tensor can be modified into a positive definite one by preserving the summations of some corresponding entries. Second, an alternating projection algorithm is proposed to verify whether an elasticity Tensor satisfies the first condition or not. Besides, the elasticity $\mathscr{M}$Tensor is defined with respect to the Meigenvalues of elasticity Tensors. We prove that any nonsingular elasticity $\mathscr{M}$Tensor satisfies the strong ellipticity condition by employing a PerronFrobeniustype theorem for Mspectral radii of nonnegative elasticity Tensors. Other equivalent definitions of nonsingular elasticity $\mathscr{M}$Tensors are also established.
Samuel Kaski  One of the best experts on this subject based on the ideXlab platform.

Bayesian multiTensor factorization
Machine Learning, 2016CoAuthors: Suleiman A. Khan, Eemeli Leppäaho, Samuel KaskiAbstract:We introduce Bayesian multiTensor factorization, a model that is the first Bayesian formulation for joint factorization of multiple matrices and Tensors. The research problem generalizes the joint matrix–Tensor factorization problem to arbitrary sets of Tensors of any depth, including matrices, can be interpreted as unsupervised multiview learning from multiple data Tensors, and can be generalized to relax the usual trilinear Tensor factorization assumptions. The result is a factorization of the set of Tensors into factors shared by any subsets of the Tensors, and factors private to individual Tensors. We demonstrate the performance against existing baselines in multiple Tensor factorization tasks in structural toxicogenomics and functional neuroimaging.