In this paper,we first initialize the S-product of tensors to unify the outer product,contractive product,and the inner product of tensors.Then,we introduce the separable symmetry tensors and separable anti-symmetry t...In this paper,we first initialize the S-product of tensors to unify the outer product,contractive product,and the inner product of tensors.Then,we introduce the separable symmetry tensors and separable anti-symmetry tensors,which are defined,respectively,as the sum and the algebraic sum of rank-one tensors generated by the tensor product of some vectors.We offer a class of tensors to achieve the upper bound for rank(A)≤6 for all tensors of size 3×3×3.We also show that each 3×3×3 anti-symmetric tensor is separable.展开更多
In this paper,we consider the positive definiteness of fourth-order partially symmetric tensors.First,two analytically sufficient and necessary conditions of positive definiteness are provided for fourth-order two dim...In this paper,we consider the positive definiteness of fourth-order partially symmetric tensors.First,two analytically sufficient and necessary conditions of positive definiteness are provided for fourth-order two dimensional partially symmetric tensors.Then,we obtain several sufficient conditions for rank-one positive definiteness of fourth-order three dimensional partially symmetric tensors.展开更多
StrongH-tensors play a significant role in identifying the positive definiteness of an even-order real symmetric tensor.In this paper,first,an improved iterative algorithm is proposed to determine whether a given tens...StrongH-tensors play a significant role in identifying the positive definiteness of an even-order real symmetric tensor.In this paper,first,an improved iterative algorithm is proposed to determine whether a given tensor is a strong H-tensor,and the validity of the iterative algorithm is proved theoretically.Second,the iterative algorithm is employed to identify the positive definiteness of an even-order real symmetric tensor.Finally,numerical examples are presented to illustrate the advantages of the proposed algorithm.展开更多
The positive definiteness of elasticity tensors plays an important role in the elasticity theory.In this paper,we consider the bi-block symmetric tensors,which contain elasticity tensors as a subclass.First,we define ...The positive definiteness of elasticity tensors plays an important role in the elasticity theory.In this paper,we consider the bi-block symmetric tensors,which contain elasticity tensors as a subclass.First,we define the bi-block M-eigenvalue of a bi-block symmetric tensor,and show that a bi-block symmetric tensor is bi-block positive(semi)definite if and only if its smallest bi-block M-eigenvalue is(nonnegative)positive.Then,we discuss the distribution of bi-block M-eigenvalues,by which we get a sufficient condition for judging bi-block positive(semi)definiteness of the bi-block symmetric tensor involved.Particularly,we show that several classes of bi-block symmetric tensors are bi-block positive definite or bi-block positive semidefinite,including bi-block(strictly)diagonally dominant symmetric tensors and bi-block symmetric(B)B0-tensors.These give easily checkable sufficient conditions for judging bi-block positive(semi)definiteness of a bi-block symmetric tensor.As a byproduct,we also obtain two easily checkable sufficient conditions for the strong ellipticity of elasticity tensors.展开更多
We study symmetric tensor spaces and cones arising from polynomial optimization and physical sciences.We prove a decomposition invariance theorem for linear operators over the symmetric tensor space,which leads to sev...We study symmetric tensor spaces and cones arising from polynomial optimization and physical sciences.We prove a decomposition invariance theorem for linear operators over the symmetric tensor space,which leads to several other interesting properties in symmetric tensor spaces.We then consider the positive semidefiniteness of linear operators which deduces the convexity of the Frobenius norm function of a symmetric tensor.Furthermore,we characterize the symmetric positive semidefinite tensor(SDT)cone by employing the properties of linear operators,design some face structures of its dual cone,and analyze its relationship to many other tensor cones.In particular,we show that the cone is self-dual if and only if the polynomial is quadratic,give specific characterizations of tensors that are in the primal cone but not in the dual for higher order cases,and develop a complete relationship map among the tensor cones appeared in the literature.展开更多
For symmetric tensors,computing generalized eigenvalues is equivalent to a homogenous polynomial optimization over the unit sphere.In this paper,we present an adaptive trustregion method for generalized eigenvalues of...For symmetric tensors,computing generalized eigenvalues is equivalent to a homogenous polynomial optimization over the unit sphere.In this paper,we present an adaptive trustregion method for generalized eigenvalues of symmetric tensors.One of the features is that the trust-region radius is automatically updated by the adaptive technique to improve the algorithm performance.The other one is that a projection scheme is used to ensure the feasibility of all iteratives.Global convergence and local quadratic convergence of our algorithm are established,respectively.The preliminary numerical results show the efficiency of the proposed algorithm.展开更多
An algorithm is presented for decomposing a symmetric tensor into a sum of rank-1 symmetric tensors. For a given tensor, by using apolarity, catalecticant matrices and the condition that the mapping matrices are commu...An algorithm is presented for decomposing a symmetric tensor into a sum of rank-1 symmetric tensors. For a given tensor, by using apolarity, catalecticant matrices and the condition that the mapping matrices are commutative, the rank of the tensor can be obtained by iteration. Then we can find the generating polynomials under a selected basis set. The decomposition can be constructed by the solutions of generating polynomials under the condition that the solutions are all distinct which can be guaranteed by the commutative property of the matrices. Numerical examples demonstrate the efficiency and accuracy of the proposed method.展开更多
We show that a best rank one approximation to a real symmetric tensor, which in principle can be nonsymmetric, can be chosen symmetric. Furthermore, a symmetric best rank one approximation to a symmetric tensor is uni...We show that a best rank one approximation to a real symmetric tensor, which in principle can be nonsymmetric, can be chosen symmetric. Furthermore, a symmetric best rank one approximation to a symmetric tensor is unique if the tensor does not lie on a certain real algebraic variety.展开更多
Symmetric tensor decomposition is of great importance in applications.Several studies have employed a greedy approach,where the main idea is to first find a best rank-one approximation of a given tensor,and then repea...Symmetric tensor decomposition is of great importance in applications.Several studies have employed a greedy approach,where the main idea is to first find a best rank-one approximation of a given tensor,and then repeat the process to the residual tensor by subtracting the rank-one component.In this paper,we focus on finding a best rank-one approximation of a given orthogonally order-3 symmetric tensor.We give a geometric landscape analysis of a nonconvex optimization for the best rank-one approximation of orthogonally symmetric tensors.We show that any local minimizer must be a factor in this orthogonally symmetric tensor decomposition,and any other critical points are linear combinations of the factors.Then,we propose a gradient descent algorithm with a carefully designed initialization to solve this nonconvex optimization problem,and we prove that the algorithm converges to the global minimum with high probability for orthogonal decomposable tensors.This result,combined with the landscape analysis,reveals that the greedy algorithm will get the tensor CP low-rank decomposition.Numerical results are provided to verify our theoretical results.展开更多
The purpose of this note is to give a linear algebra algorithm to find out if a rank of a given tensor over a field F is at most k over the algebraic closure of F,where K is a given positive integer.We estimate the ar...The purpose of this note is to give a linear algebra algorithm to find out if a rank of a given tensor over a field F is at most k over the algebraic closure of F,where K is a given positive integer.We estimate the arithmetic complexity of our algorithm.展开更多
The main purpose of this paper is to consider the Perron pair of an irreducible and symmetric nonnegative tensor and the smallest eigenvalue of an irreducible and symmetric nonsingular M-tensor.We analyze the analytic...The main purpose of this paper is to consider the Perron pair of an irreducible and symmetric nonnegative tensor and the smallest eigenvalue of an irreducible and symmetric nonsingular M-tensor.We analyze the analytical property of an algebraic simple eigenvalue of symmetric tensors.We also derive an inequality about the Perron pair of nonnegative tensors based on plane stochastic tensors.We finally consider the perturbation of the smallest eigenvalue of nonsingular M-tensors and design a strategy to compute its smallest eigenvalue.We verify our results via random numerical examples.展开更多
In this paper some properties of a symmetric tensor field T（X,Y） = g（A（X）, Y） on a Riemannian manifold （M, g） without boundary which satisfies the S quasi-Einstein equation Rij-S/2gij=Tij＋bξiξj are given. ...In this paper some properties of a symmetric tensor field T（X,Y） = g（A（X）, Y） on a Riemannian manifold （M, g） without boundary which satisfies the S quasi-Einstein equation Rij-S/2gij=Tij＋bξiξj are given. The necessary and sufficient conditions for this tensor to satisfy the quasi-Einstein equation are also obtained.展开更多
Some new criteria for identifying H-tensors are obtained. As applications, some sufficient conditions of the positive definiteness for an even- order real symmetric tensor are given, as well as a new eigenvalue inclus...Some new criteria for identifying H-tensors are obtained. As applications, some sufficient conditions of the positive definiteness for an even- order real symmetric tensor are given, as well as a new eigenvalue inclusion region for tensors is established. It is proved that the new eigenvalue inclusion region is tighter than that of Y. Yang and Q. Yang [SIAM J. Matrix Anal. Appl., 2010, 31： 2517-2530]. Numerical examples are reported to demonstrate the corresponding results.展开更多
The problem of computing geometric measure of quantum entanglement for symmetric pure states can be regarded as the problem of finding the largest unitary symmetric eigenvalue(US-eigenvalue)for symmetric complex tenso...The problem of computing geometric measure of quantum entanglement for symmetric pure states can be regarded as the problem of finding the largest unitary symmetric eigenvalue(US-eigenvalue)for symmetric complex tensors,which can be taken as a multilinear optimization problem in complex number field.In this paper,we convert the problem of computing the geometric measure of entanglement for symmetric pure states to a real polynomial optimization problem.Then we use Jacobian semidefinite relaxation method to solve it.Some numerical examples are presented.展开更多
Let A be an ruth order n-dimensional tensor, where m, n are some positive integers and N ：= re（n-1）. Then A is called a Hankel tensor associated with a vector v ∈ R^N＋1 if Aσ = Vk for each k = 0,1,...,N whenever...Let A be an ruth order n-dimensional tensor, where m, n are some positive integers and N ：= re（n-1）. Then A is called a Hankel tensor associated with a vector v ∈ R^N＋1 if Aσ = Vk for each k = 0,1,...,N whenever σ= （i1,..., im） satisfies i1 ＋... ＋ im - m ＋ k. We introduce the elementary Hankel tensors which are some special Hankel tensors, and present all the eigenvalues of the elementary Hankel tensors for k = 0, 1, 2. We also show that a convolution can be expressed as the product of some third-order elementary Hankel tensors, and a Hankel tensor can be decomposed as a convolution of two Vandermonde matrices following the definition of the convolution of tensors. Finally, we use the properties of the convolution to characterize Hankel tensors and （0,1） Hankel tensors. Keywords Tensor, convolution, Hankel tensor, elementary Hankel tensor, symmetric tensor展开更多
Based on the generalized characteristic polynomial introduced by J. Canny in Generalized characteristic polynomials [J. Symbolic Comput., 1990, 9（3）： 241-250], it is immediate that for any m-order n-dimensional rea...Based on the generalized characteristic polynomial introduced by J. Canny in Generalized characteristic polynomials [J. Symbolic Comput., 1990, 9（3）： 241-250], it is immediate that for any m-order n-dimensional real tensor, the number of distinct H-eigenvalues is less than or equal to n（m-1）n-1. However, there is no known bounds on the maximal number of distinct H- eigenvectors in general. We prove that for any m ~〉 2, an m-order 2-dimensional tensor sd exists such that d has 2（m - 1） distinct H-eigenpairs. We give examples of 4-order 2-dimensional tensors with six distinct H-eigenvalues as well as six distinct H-eigenvectors. We demonstrate the structure of eigenpairs for a higher order tensor is far more complicated than that of a matrix. Further- more, we introduce a new class of weakly symmetric tensors, called p-symmetric tensors, and show under certain conditions, p-symmetry will effectively reduce the maximal number of distinct H-eigenveetors for a given two-dimensional tensor. Lastly, we provide a complete classification of the H-eigenvectors of a given 4-order 2-dimensional nonnegative p-symmetric tensor. Additionally, we give sufficient conditions which prevent a given 4-order 2-dimensional nonnegative irreducible weakly symmetric tensor from possessing six pairwise distinct H-eigenveetors.展开更多
This paper introduces a new family of mixed finite elements for solving a mixed formulation of the biharmonic equations in two and three dimensions.The symmetric stress σ=−∇^(2)u is sought in the Sobolev space H(divd...This paper introduces a new family of mixed finite elements for solving a mixed formulation of the biharmonic equations in two and three dimensions.The symmetric stress σ=−∇^(2)u is sought in the Sobolev space H(divdiv,Ω;S)simultaneously with the displacement u in L^(2)(Ω).By stemming from the structure of H(div,Ω;S)conforming elements for the linear elasticity problems proposed by Hu and Zhang(2014),the H(divdiv,Ω;S)conforming finite element spaces are constructed by imposing the normal continuity of divσ on the H(div,Ω;S)conforming spaces of P_(k) symmetric tensors.The inheritance makes the basis functions easy to compute.The discrete spaces for u are composed of the piecewise P_(k−2) polynomials without requiring any continuity.Such mixed finite elements are inf-sup stable on both triangular and tetrahedral grids for k≥3,and the optimal order of convergence is achieved.Besides,the superconvergence and the postprocessing results are displayed.Some numerical experiments are provided to demonstrate the theoretical analysis.展开更多
文摘In this paper,we first initialize the S-product of tensors to unify the outer product,contractive product,and the inner product of tensors.Then,we introduce the separable symmetry tensors and separable anti-symmetry tensors,which are defined,respectively,as the sum and the algebraic sum of rank-one tensors generated by the tensor product of some vectors.We offer a class of tensors to achieve the upper bound for rank(A)≤6 for all tensors of size 3×3×3.We also show that each 3×3×3 anti-symmetric tensor is separable.
文摘In this paper,we consider the positive definiteness of fourth-order partially symmetric tensors.First,two analytically sufficient and necessary conditions of positive definiteness are provided for fourth-order two dimensional partially symmetric tensors.Then,we obtain several sufficient conditions for rank-one positive definiteness of fourth-order three dimensional partially symmetric tensors.
文摘StrongH-tensors play a significant role in identifying the positive definiteness of an even-order real symmetric tensor.In this paper,first,an improved iterative algorithm is proposed to determine whether a given tensor is a strong H-tensor,and the validity of the iterative algorithm is proved theoretically.Second,the iterative algorithm is employed to identify the positive definiteness of an even-order real symmetric tensor.Finally,numerical examples are presented to illustrate the advantages of the proposed algorithm.
基金The first author’s work was supported by the National Natural Science Foundation of China(Grant No.11871051).
文摘The positive definiteness of elasticity tensors plays an important role in the elasticity theory.In this paper,we consider the bi-block symmetric tensors,which contain elasticity tensors as a subclass.First,we define the bi-block M-eigenvalue of a bi-block symmetric tensor,and show that a bi-block symmetric tensor is bi-block positive(semi)definite if and only if its smallest bi-block M-eigenvalue is(nonnegative)positive.Then,we discuss the distribution of bi-block M-eigenvalues,by which we get a sufficient condition for judging bi-block positive(semi)definiteness of the bi-block symmetric tensor involved.Particularly,we show that several classes of bi-block symmetric tensors are bi-block positive definite or bi-block positive semidefinite,including bi-block(strictly)diagonally dominant symmetric tensors and bi-block symmetric(B)B0-tensors.These give easily checkable sufficient conditions for judging bi-block positive(semi)definiteness of a bi-block symmetric tensor.As a byproduct,we also obtain two easily checkable sufficient conditions for the strong ellipticity of elasticity tensors.
基金supported by National Natural Science Foundation of China(Grant No.11301022)the State Key Laboratory of Rail Traffic Control and Safety,Beijing Jiaotong University(Grant Nos.RCS2014ZT20 and RCS2014ZZ001)+1 种基金Beijing Natural Science Foundation(Grant No.9144031)the Hong Kong Research Grant Council(Grant Nos.Poly U 501909,502510,502111 and 501212)
文摘We study symmetric tensor spaces and cones arising from polynomial optimization and physical sciences.We prove a decomposition invariance theorem for linear operators over the symmetric tensor space,which leads to several other interesting properties in symmetric tensor spaces.We then consider the positive semidefiniteness of linear operators which deduces the convexity of the Frobenius norm function of a symmetric tensor.Furthermore,we characterize the symmetric positive semidefinite tensor(SDT)cone by employing the properties of linear operators,design some face structures of its dual cone,and analyze its relationship to many other tensor cones.In particular,we show that the cone is self-dual if and only if the polynomial is quadratic,give specific characterizations of tensors that are in the primal cone but not in the dual for higher order cases,and develop a complete relationship map among the tensor cones appeared in the literature.
基金supported in part by the NNSF of China(11171003)the Innovation Talent Training Program of Science and Technology of Jilin Province of China(20180519011JH)+2 种基金the Science and Technology Development Project Program of Jilin Province(20190303132SF)The research of Mingyuan Cao is partially supported by the Project of Education Department of Jilin Province(JJKH20200028KJ)The research of Qingdao Huang is partially supported by the NNSF of China(11171131).
文摘For symmetric tensors,computing generalized eigenvalues is equivalent to a homogenous polynomial optimization over the unit sphere.In this paper,we present an adaptive trustregion method for generalized eigenvalues of symmetric tensors.One of the features is that the trust-region radius is automatically updated by the adaptive technique to improve the algorithm performance.The other one is that a projection scheme is used to ensure the feasibility of all iteratives.Global convergence and local quadratic convergence of our algorithm are established,respectively.The preliminary numerical results show the efficiency of the proposed algorithm.
基金This work was supported by the National Natural Science Foundation of China （Grants Nos. 11471159, 11571169, 61661136001） and the Natural Science Foundation of Jiangsu Province （No. BK20141409）.
文摘An algorithm is presented for decomposing a symmetric tensor into a sum of rank-1 symmetric tensors. For a given tensor, by using apolarity, catalecticant matrices and the condition that the mapping matrices are commutative, the rank of the tensor can be obtained by iteration. Then we can find the generating polynomials under a selected basis set. The decomposition can be constructed by the solutions of generating polynomials under the condition that the solutions are all distinct which can be guaranteed by the commutative property of the matrices. Numerical examples demonstrate the efficiency and accuracy of the proposed method.
文摘We show that a best rank one approximation to a real symmetric tensor, which in principle can be nonsymmetric, can be chosen symmetric. Furthermore, a symmetric best rank one approximation to a symmetric tensor is unique if the tensor does not lie on a certain real algebraic variety.
文摘Symmetric tensor decomposition is of great importance in applications.Several studies have employed a greedy approach,where the main idea is to first find a best rank-one approximation of a given tensor,and then repeat the process to the residual tensor by subtracting the rank-one component.In this paper,we focus on finding a best rank-one approximation of a given orthogonally order-3 symmetric tensor.We give a geometric landscape analysis of a nonconvex optimization for the best rank-one approximation of orthogonally symmetric tensors.We show that any local minimizer must be a factor in this orthogonally symmetric tensor decomposition,and any other critical points are linear combinations of the factors.Then,we propose a gradient descent algorithm with a carefully designed initialization to solve this nonconvex optimization problem,and we prove that the algorithm converges to the global minimum with high probability for orthogonal decomposable tensors.This result,combined with the landscape analysis,reveals that the greedy algorithm will get the tensor CP low-rank decomposition.Numerical results are provided to verify our theoretical results.
文摘The purpose of this note is to give a linear algebra algorithm to find out if a rank of a given tensor over a field F is at most k over the algebraic closure of F,where K is a given positive integer.We estimate the arithmetic complexity of our algorithm.
基金the National Natural Science Foundation of China(No.11271084)International Cooperation Project of Shanghai Municipal Science and Technology Commission(No.16510711200).
文摘The main purpose of this paper is to consider the Perron pair of an irreducible and symmetric nonnegative tensor and the smallest eigenvalue of an irreducible and symmetric nonsingular M-tensor.We analyze the analytical property of an algebraic simple eigenvalue of symmetric tensors.We also derive an inequality about the Perron pair of nonnegative tensors based on plane stochastic tensors.We finally consider the perturbation of the smallest eigenvalue of nonsingular M-tensors and design a strategy to compute its smallest eigenvalue.We verify our results via random numerical examples.
基金The Grant-in-Aid for Scientific Research from Nanjing University of ScienceTechnology (AB41409) the NNSF (19771048) of China partly.
文摘In this paper some properties of a symmetric tensor field T（X,Y） = g（A（X）, Y） on a Riemannian manifold （M, g） without boundary which satisfies the S quasi-Einstein equation Rij-S/2gij=Tij＋bξiξj are given. The necessary and sufficient conditions for this tensor to satisfy the quasi-Einstein equation are also obtained.
基金Acknowledgements This work was supported by the National Natural Science Foundation of China （Grant Nos. 11361074, 11326242） and the Science Foundation of the Education Department of Yunnan Province （Grant No. 2013FD002）.
文摘Some new criteria for identifying H-tensors are obtained. As applications, some sufficient conditions of the positive definiteness for an even- order real symmetric tensor are given, as well as a new eigenvalue inclusion region for tensors is established. It is proved that the new eigenvalue inclusion region is tighter than that of Y. Yang and Q. Yang [SIAM J. Matrix Anal. Appl., 2010, 31： 2517-2530]. Numerical examples are reported to demonstrate the corresponding results.
基金the Research Programme of National University of Defense Technology(No.ZK16-03-45).
文摘The problem of computing geometric measure of quantum entanglement for symmetric pure states can be regarded as the problem of finding the largest unitary symmetric eigenvalue(US-eigenvalue)for symmetric complex tensors,which can be taken as a multilinear optimization problem in complex number field.In this paper,we convert the problem of computing the geometric measure of entanglement for symmetric pure states to a real polynomial optimization problem.Then we use Jacobian semidefinite relaxation method to solve it.Some numerical examples are presented.
文摘Let A be an ruth order n-dimensional tensor, where m, n are some positive integers and N ：= re（n-1）. Then A is called a Hankel tensor associated with a vector v ∈ R^N＋1 if Aσ = Vk for each k = 0,1,...,N whenever σ= （i1,..., im） satisfies i1 ＋... ＋ im - m ＋ k. We introduce the elementary Hankel tensors which are some special Hankel tensors, and present all the eigenvalues of the elementary Hankel tensors for k = 0, 1, 2. We also show that a convolution can be expressed as the product of some third-order elementary Hankel tensors, and a Hankel tensor can be decomposed as a convolution of two Vandermonde matrices following the definition of the convolution of tensors. Finally, we use the properties of the convolution to characterize Hankel tensors and （0,1） Hankel tensors. Keywords Tensor, convolution, Hankel tensor, elementary Hankel tensor, symmetric tensor
文摘Based on the generalized characteristic polynomial introduced by J. Canny in Generalized characteristic polynomials [J. Symbolic Comput., 1990, 9（3）： 241-250], it is immediate that for any m-order n-dimensional real tensor, the number of distinct H-eigenvalues is less than or equal to n（m-1）n-1. However, there is no known bounds on the maximal number of distinct H- eigenvectors in general. We prove that for any m ~〉 2, an m-order 2-dimensional tensor sd exists such that d has 2（m - 1） distinct H-eigenpairs. We give examples of 4-order 2-dimensional tensors with six distinct H-eigenvalues as well as six distinct H-eigenvectors. We demonstrate the structure of eigenpairs for a higher order tensor is far more complicated than that of a matrix. Further- more, we introduce a new class of weakly symmetric tensors, called p-symmetric tensors, and show under certain conditions, p-symmetry will effectively reduce the maximal number of distinct H-eigenveetors for a given two-dimensional tensor. Lastly, we provide a complete classification of the H-eigenvectors of a given 4-order 2-dimensional nonnegative p-symmetric tensor. Additionally, we give sufficient conditions which prevent a given 4-order 2-dimensional nonnegative irreducible weakly symmetric tensor from possessing six pairwise distinct H-eigenveetors.
基金supported by National Natural Science Foundation of China(Grant Nos.11625101 and 11421101).
文摘This paper introduces a new family of mixed finite elements for solving a mixed formulation of the biharmonic equations in two and three dimensions.The symmetric stress σ=−∇^(2)u is sought in the Sobolev space H(divdiv,Ω;S)simultaneously with the displacement u in L^(2)(Ω).By stemming from the structure of H(div,Ω;S)conforming elements for the linear elasticity problems proposed by Hu and Zhang(2014),the H(divdiv,Ω;S)conforming finite element spaces are constructed by imposing the normal continuity of divσ on the H(div,Ω;S)conforming spaces of P_(k) symmetric tensors.The inheritance makes the basis functions easy to compute.The discrete spaces for u are composed of the piecewise P_(k−2) polynomials without requiring any continuity.Such mixed finite elements are inf-sup stable on both triangular and tetrahedral grids for k≥3,and the optimal order of convergence is achieved.Besides,the superconvergence and the postprocessing results are displayed.Some numerical experiments are provided to demonstrate the theoretical analysis.
