Strictly nonnegative tensors and nonnegative tensor partition

We introduce a new class of nonnegative tensors—strictly nonnegative tensors.A weakly irreducible nonnegative tensor is a strictly nonnegative tensor but not vice versa.We show that the spectral radius of a strictly nonnegative tensor is always positive.We give some necessary and su?cient conditions for the six wellconditional classes of nonnegative tensors,introduced in the literature,and a full relationship picture about strictly nonnegative tensors with these six classes of nonnegative tensors.We then establish global R-linear convergence of a power method for finding the spectral radius of a nonnegative tensor under the condition of weak irreducibility.We show that for a nonnegative tensor T,there always exists a partition of the index set such that every tensor induced by the partition is weakly irreducible;and the spectral radius of T can be obtained from those spectral radii of the induced tensors.In this way,we develop a convergent algorithm for finding the spectral radius of a general nonnegative tensor without any additional assumption.Some preliminary numerical results show the feasibility and effectiveness of the algorithm.     

Singular values of nonnegative rectangular tensors

The real rectangular tensors arise from the strong ellipticity condition problem in solid mechanics and the entanglement problem in quantum physics. Some properties concerning the singular values of a real rectangular tensor were discussed by K. C. Chang et al. [J. Math. Anal. Appl., 2010, 370: 284–294]. In this paper, we give some new results on the Perron-Frobenius Theorem for nonnegative rectangular tensors. We show that the weak Perron-Frobenius keeps valid and the largest singular value is really geometrically simple under some conditions. In addition, we establish the convergence of an algorithm proposed by K. C. Chang et al. for finding the largest singular value of nonnegative primitive rectangular tensors.     

A semidefinite program approach for computing the maximum eigenvalue of a class of structured tensors and its applications in hypergraphs and copositivity test

Finding the maximum eigenvalue of a symmetric tensor is an important topic in tensor computation and numerical multilinear algebra. In this paper, we introduce a new class of structured tensors called W‐tensors, which not only extends the well‐studied nonnegative tensors by allowing negative entries but also covers several important tensors arising naturally from spectral hypergraph theory. We then show that finding the maximum H‐eigenvalue of an even‐order symmetric W‐tensor is equivalent to solving a structured semidefinite program and hence can be validated in polynomial time. This yields a highly efficient semidefinite program algorithm for computing the maximum H‐eigenvalue of W‐tensors and is based on a new structured sums‐of‐squares decomposition result for a nonnegative polynomial induced by W‐tensors. Numerical experiments illustrate that the proposed algorithm can successfully find the maximum H‐eigenvalue of W‐tensors with dimension up to 10,000, subject to machine precision. As applications, we provide a polynomial time algorithm for computing the maximum H‐eigenvalues of large‐size Laplacian tensors of hyperstars and hypertrees, where the algorithm can be up to 13 times faster than the state‐of‐the‐art numerical method introduced by Ng, Qi, and Zhou in 2009. Finally, we also show that the proposed algorithm can be used to test the copositivity of a multivariate form associated with symmetric extended Z‐tensors, whose order may be even or odd.     

Column sufficient tensors and tensor complementarity problems

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 semi-definite 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 H-eigenvalues of an even-order symmetric column sufficient tensor are nonnegative, and all its Z-eigenvalues 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.     

New eigenvalue inclusion sets for tensors

Two new eigenvalue inclusion sets for tensors are established. It is proved that the new eigenvalue inclusion sets are tighter than that in Qi's paper “Eigenvalues of a real supersymmetric tensor”. As applications, upper bounds for the spectral radius of a nonnegative tensor are obtained, and it is proved that these upper bounds are sharper than that in Yang's paper “Further results for Perron–Frobenius theorem for nonnegative tensors”. And some sufficient conditions of the positive definiteness for an even‐order real supersymmetric tensor are given. Copyright © 2012 John Wiley & Sons, Ltd.     

Geometric simplicity of spectral radius of nonnegative irreducible tensors

We study the real and complex geometric simplicity of nonnegative irreducible tensors. First, we prove some basic conclusions. Based on the conclusions, the real geometric simplicity of the spectral radius of an evenorder nonnegative irreducible tensor is proved. For an odd-order nonnegative irreducible tensor, sufficient conditions are investigated to ensure the spectral radius to be real geometrically simple. Furthermore, the complex geometric simplicity of nonnegative irreducible tensors is also studied.     

A fast algorithm for the spectral radii of weakly reducible nonnegative tensors

In this paper, we propose a fast algorithm for computing the spectral radii of symmetric nonnegative tensors. In particular, by this proposed algorithm, we are able to obtain the spectral radii of weakly reducible symmetric nonnegative tensors without requiring the partition of the tensors. As we know, it is very costly to determine the partition for large‐sized weakly reducible tensors. Numerical results are reported to show that the proposed algorithm is efficient and also able to compute the spectral radii of large‐sized tensors. As an application, we present an algorithm for testing the positive definiteness of Z‐tensors. By this algorithm, it is guaranteed to determine the positive definiteness for any Z‐tensor.     

Standard tensor and its applications in problem of singular values of tensors

In this paper, first we give the definition of standard tensor. Then we clarify the relationship between weakly irreducible tensors and weakly irreducible polynomial maps by the definition of standard tensor. And we prove that the singular values of rectangular tensors are the special cases of the eigen-values of standard tensors related to rectangular tensors. Based on standard tensor, we present a generalized version of the weak Perron-Frobenius Theorem of nonnegative rectangular tensors under weaker conditions. Furthermore, by studying standard tensors, we get some new results of rectangular tensors. Besides, by using the special structure of standard tensors corresponding to nonnegative rectangular tensors, we show that the largest singular value is really geometrically simple under some weaker conditions.     

Bi-block positive semidefiniteness of bi-block symmetric tensors

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.     

Generalized inverses of tensors via a general product of tensors

We define the {i}-inverse (i = 1, 2, 5) and group inverse of tensors based on a general product of tensors. We explore properties of the generalized inverses of tensors on solving tensor equations and computing formulas of block tensors. We use the {1}-inverse of tensors to give the solutions of a multilinear system represented by tensors. The representations for the {1}-inverse and group inverse of some block tensors are established.     

Singular value inclusion sets for rectangular tensors

Two singular value inclusion sets for rectangular tensors are given. These sets provide two upper bounds and lower bounds for the largest singular value of nonnegative rectangular tensors, which can be taken as a parameter of an algorithm presented by Zhou et al. (Linear Algebra Appl. 2013; 438: 959–968) such that the sequences produced by this algorithm converge rapidly to the largest singular value of an irreducible nonnegative rectangular tensor.     

l k,s -Singular values and spectral radius of rectangular tensors

The real rectangular tensors arise from the strong ellipticity condition problem in solid mechanics and the entanglement problem in quantum physics. In this paper, we study the singular values/vectors problem of real nonnegative partially symmetric rectangular tensors. We first introduce the concepts of l ^k,s -singular values/vectors of real partially symmetric rectangular tensors. Then, based upon the presented properties of l ^k,s -singular values /vectors, some properties of the related l ^k,s -spectral radius are discussed. Furthermore, we prove two analogs of Perron-Frobenius theorem and weak Perron-Frobenius theorem for real nonnegative partially symmetric rectangular tensors.     

On computing minimal <Emphasis Type="Italic">H</Emphasis>-eigenvalue of sign-structured tensors

Finding the minimal H-eigenvalue of tensors is an important topic in tensor computation and numerical multilinear algebra. This paper is devoted to a sum-of-squares (SOS) algorithm for computing the minimal H-eigenvalues of tensors with some sign structures called extended essentially nonnegative tensors (EEN-tensors), which includes nonnegative tensors as a subclass. In the even-order symmetric case, we first discuss the positive semi-definiteness of EEN-tensors, and show that a positive semi-definite EEN-tensor is a nonnegative tensor or an M-tensor or the sum of a nonnegative tensor and an M-tensor, then we establish a checkable sufficient condition for the SOS decomposition of EEN-tensors. Finally, we present an efficient algorithm to compute the minimal H-eigenvalues of even-order symmetric EEN-tensors based on the SOS decomposition. Numerical experiments are given to show the efficiency of the proposed algorithm.     

A new definition of geometric multiplicity of eigenvalues of tensors and some results based on it

We give a new definition of geometric multiplicity of eigenvalues of tensors, and based on this, we study the geometric and algebraic multiplicity of irreducible tensors’ eigenvalues. We get the result that the eigenvalues with modulus ρ(A) have the same geometric multiplicity. We also prove that twodimensional nonnegative tensors’ geometric multiplicity of eigenvalues is equal to algebraic multiplicity of eigenvalues.     

Reconstruction of convex bodies from surface tensors

We present two algorithms for reconstruction of the shape of convex bodies in the two-dimensional Euclidean space. The first reconstruction algorithm requires knowledge of the exact surface tensors of a convex body up to rank s for some natural number s. When only measurements subject to noise of surface tensors are available for reconstruction, we recommend to use certain values of the surface tensors, namely harmonic intrinsic volumes instead of the surface tensors evaluated at the standard basis. The second algorithm we present is based on harmonic intrinsic volumes and allows for noisy measurements. From a generalized version of Wirtinger's inequality, we derive stability results that are utilized to ensure consistency of both reconstruction procedures. Consistency of the reconstruction procedure based on measurements subject to noise is established under certain assumptions on the noise variables.     

Efficient algorithms for computing the largest eigenvalue of a nonnegative tensor

Consider the problem of computing the largest eigenvalue for nonnegative tensors. In this paper, we establish the Q-linear convergence of a power type algorithm for this problem under a weak irreducibility condition. Moreover, we present a convergent algorithm for calculating the largest eigenvalue for any nonnegative tensors.     

Brauer-type bounds for Hadamard product of nonnegative tensors

In this paper, we establish some Brauer-type bounds for the spectral radius of Hadamard product of two nonnegative tensors based on Brauer-type inclusion set, which are shown to be sharper than the existing bounds established in the literature. The validity of the obtained results is theoretically and numerically tested.     

Biquadratic tensors,biquadratic decompositions,and norms of biquadratic tensors

Biquadratic tensors play a central role in many areas of science.Examples include elastic tensor and Eshelby tensor in solid mechanics,and Riemannian curvature tensor in relativity theory.The singular values and spectral norm of a general third order tensor are the square roots of the M-eigenvalues and spectral norm of a biquadratic tensor,respectively.The tensor product operation is closed for biquadratic tensors.All of these motivate us to study biquadratic tensors,biquadratic decomposition,and norms of biquadratic tensors.We show that the spectral norm and nuclear norm for a biquadratic tensor may be computed by using its biquadratic structure.Then,either the number of variables is reduced,or the feasible region can be reduced.We show constructively that for a biquadratic tensor,a biquadratic rank-one decomposition always exists,and show that the biquadratic rank of a biquadratic tensor is preserved under an independent biquadratic Tucker decomposition.We present a lower bound and an upper bound of the nuclear norm of a biquadratic tensor.Finally,we define invertible biquadratic tensors,and present a lower bound for the product of the nuclear norms of an invertible biquadratic tensor and its inverse,and a lower bound for the product of the nuclear norm of an invertible biquadratic tensor,and the spectral norm of its inverse.     

Linear convergence of an algorithm for computing the largest eigenvalue of a nonnegative tensor

An iterative method for finding the largest eigenvalue of a nonnegative tensor was proposed by Ng, Qi, and Zhou in 2009. In this paper, we establish an explicit linear convergence rate of the Ng–Qi–Zhou method for essentially positive tensors. Numerical results are given to demonstrate linear convergence of the Ng–Qi–Zhou algorithm for essentially positive tensors. Copyright © 2011 John Wiley & Sons, Ltd.     

A general product of tensors with applications

We study a general product of two n  -dimensional tensors A$\mathbb{A}$ and B$\mathbb{B}$ with orders m?2$m?2$ and k?1$k?1$. This product satisfies the associative law, and is a generalization of the usual matrix product. Using this product, many concepts and known results of tensors can be simply expressed and/or proved, and a number of applications of it will be given. Using the associative law of this tensor product and some properties on the resultant of a system of homogeneous equations on n variables, we define the similarity and congruence of tensors (which are also the generalizations of the corresponding relations for matrices), and prove that similar tensors have the same characteristic polynomials, thus the same spectra. We study two special kinds of similarity: permutational similarity and diagonal similarity, and their applications in the study of the spectra of hypergraphs and nonnegative irreducible tensors. We also define the direct product of tensors (in matrix case it is also called the Kronecker product), and give its applications in the study of the spectra of two kinds of the products of hypergraphs. We also give applications of this general product in the study of nonnegative tensors, including a characterization of primitive tensors, the upper bounds of primitive degrees and the cyclic indices of some nonnegative irreducible tensors.