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The Fourier matrix is fundamental in discrete Fourier transforms and fast Fourier transforms. We generalize the Fourier matrix, extend the concept of Fourier matrix to higher order Fourier tensor, present the spectrum of the Fourier tensors, and use the Fourier tensor to simplify the high order Fourier analysis.

We present upper bounds of eigenvalues for finite and infinite dimensional Cauchy-Hankel tensors. It is proved that an m-order infinite dimensional Cauchy-Hankel tensor defines a bounded and positively (m − 1)-homogeneous operator from l1 into lp (1 < p < ∞), and two upper bounds of corresponding positively homogeneous operator norms are given. Moreover, for a fourth-order real partially symmetric Cauchy-Hankel tensor, sufficient and necessary conditions of M-positive definiteness are obtained, and an upper bound of M-eigenvalue is also shown.

A real symmetric tensor $${\mathscr{A}} = ({a_{{i_1} \ldots {i_m}}}) \in {\mathbb{R}^{[m,n]}}$$ is copositive (resp., strictly copositive) if $${\mathscr{A}}\;{x^m} \geqslant 0$$ (resp., $${\mathscr{A}}\;{x^m} > 0$$ ) for any nonzero nonnegative vector x ∈ ℝn. By using the associated hypergraph of $${\mathscr{A}}$$ , we give necessary and sufficient conditions for the copositivity of $${\mathscr{A}}$$ . For a real symmetric tensor $${\mathscr{A}}$$ satisfying the associated negative hypergraph $${H_ - }({\mathscr{A}})$$ and associated positive hypergraph $${H_ + }({\mathscr{A}})$$ are edge disjoint subhypergraphs of a supertree or cored hypergraph, we derive criteria for the copositivity of $${\mathscr{A}}$$ . We also use copositive tensors to study the positivity of tensor systems.

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.

We establish necessary and sufficient conditions for the existence of the reducible solution to the quaternion tensor equation $$\mathscr{A}{ * _N}\mathscr{X}{ * _N}\mathscr{B} = \mathscr{C}$$ via Einstein product using Moore-Penrose inverse, and present an expression of the reducible solution to the equation when it is solvable. Moreover, to have a general solution, we give the solvability conditions for the quaternion tensor equation $${\mathscr{A}_1}{ * _N}{\mathscr{X}_1}{ * _M}{\mathscr{B}_1} + {\mathscr{A}_1}{ * _N}{\mathscr{X}_2}{ * _M}{\mathscr{B}_2} + {\mathscr{A}_2}{ * _N}{\mathscr{X}_3}{ * _M}{\mathscr{B}_2} = \mathscr{C}$$ , which plays a key role in investigating the reducible solution to $$\mathscr{A}{ * _N}\mathscr{X}{ * _N}\mathscr{B} = \mathscr{C}$$ . The expression of such a solution is also presented when the consistency conditions are met. In addition, we show a numerical example to illustrate this result.

This paper develops the Bernstein tensor concentration inequality for random tensors of general order, based on the use of Einstein products for tensors. This establishes a strong link between these and matrices, which in turn allows exploitation of existing results for the latter. An interesting application to sample estimators of high-order moments is presented as an illustration.

Tensor decomposition is an important research area with numerous applications in data mining and computational neuroscience. An important class of tensor decomposition is sum-of-squares (SOS) tensor decomposition. SOS tensor decomposition has a close connection with SOS polynomials, and SOS polynomials are very important in polynomial theory and polynomial optimization. In this paper, we give a detailed survey on recent advances of high-order SOS tensors and their applications. It first shows that several classes of symmetric structured tensors available in the literature have SOS decomposition in the even order symmetric case. Then, the SOS-rank for tensors with SOS decomposition and the SOS-width for SOS tensor cones are established. Further, a sharper explicit upper bound of the SOS-rank for tensors with bounded exponent is provided, and the exact SOS-width for the cone consists of all such tensors with SOS decomposition is identified. Some potential research directions in the future are also listed in this paper.

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.

We introduce a class of structured tensors, called generalized row strictly diagonally dominant tensors, and discuss some relationships between it and several classes of structured tensors, including nonnegative tensors, B-tensors, and strictly copositive tensors. In particular, we give estimations on upper and lower bounds of solutions to the tensor complementarity problem (TCP) when the involved tensor is a generalized row strictly diagonally dominant tensor with all positive diagonal entries. The main advantage of the results obtained in this paper is that both bounds we obtained depend only on the tensor and constant vector involved in the TCP; and hence, they are very easy to calculate.

Let ℋ be a k-uniform hypergraph on n vertices with degree sequence Δ = d1 ⩾ ⋯ ⩾ dn = δ In this paper, in terms of degree di; we give some upper bounds for the Z-spectral radius of the signless Laplacian tensor (Q(ℋ)) of ℋ. Some examples are given to show the efficiency of these bounds.

Eigenvalues of tensors play an increasingly important role in many aspects of applied mathematics. The characteristic polynomial provides one of a very few ways that shed lights on intrinsic understanding of the eigenvalues. It is known that the characteristic polynomial of a third order three dimensional tensor has a stunning expression with more than 20000 terms, thus prohibits an effective analysis. In this article, we are trying to make a concise representation of this characteristic polynomial in terms of certain basic determinants. With this, we can successfully write out explicitly the characteristic polynomial of a third order three dimensional tensor in a reasonable length. An immediate benefit is that we can compute out the third and fourth order traces of a third order three dimensional tensor symbolically, which is impossible in the literature.

We introduce two new types of tensors called the strictly semimonotone tensor and the range column sufficient tensor and explore their structure properties. Based on the obtained results, we make a characterization to the solution of tensor complementarity problem.

We discuss the uniqueness and the perturbation analysis for sparse non-negative tensor equations arriving from data sciences. By two different techniques, we may get better ranges of parameters to guarantee the uniqueness of the solution of the tensor equation. On the other hand, we present some perturbation bounds for the tensor equation. Numerical examples are given to show the efficiency of the theoretical results.

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.

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.

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.

We treat even-order tensors with Einstein product as linear operators from tensor space to tensor space, define the null spaces and the ranges of tensors, and study their relationship. We extend the fundamental theorem of linear algebra for matrix spaces to tensor spaces. Using the new relationship, we characterize the least-squares (M) solutions to a multilinear system and establish the relationship between the minimum-norm (N) least-squares (M) solution of a multilinear system and the weighted Moore-Penrose inverse of its coefficient tensor. We also investigate a class of even-order tensors induced by matrices and obtain some interesting properties.

We give a further study on B-tensors and introduce doubly Btensors that contain B-tensors. We show that they have similar properties, including their decompositions and strong relationship with strictly (doubly) diagonally dominated tensors. As an application, the properties of B-tensors are used to localize real eigenvalues of some tensors, which would be very useful in verifying the positive semi-definiteness of a tensor.

H-tensor is a new developed concept which plays an important role in tensor analysis and computing. In this paper, we explore the properties of H-tensors and establish some new criteria for strong H-tensors. In particular, based on the principal subtensor, we provide a new necessary and sufficient condition of strong H-tensors, and based on a type of generalized diagonal product dominance, we establish some new criteria for identifying strong H-tensors. The results obtained in this paper extend the corresponding conclusions for strong H-matrices and improve the existing results for strong H-tensors.

We extend Vandermonde matrices to generalized Vandermonde tensors. We call an mth order n-dimensional real tensor $$A = (A_{i_1 i_2 } \cdots _{i_m } )$$ a type-1 generalized Vandermonde (GV) tensor, or GV 1 tensor, if there exists a vector v = (v 1, v 2,..., v n )T such that $${A_{{i_1},{i_2} \ldots {i_m}}} = v_{{i_1}}^{{i_2} + {i_3} + \ldots + {i_m} - m + 1},$$ and call A a type-2 (mth order n dimensional) GV tensor, or GV 2 tensor, if there exists an (m − 1)th order tensor $$B = \left( {{B_{{i_1}{i_2} \ldots {i_{m - 1}}}}} \right)$$ such that $${A_{{i_1},{i_2} \ldots {i_m}}} = B_{{i_1}{i_{2 \ldots {i_{m - 1}}}}}^{{i_m} - 1},$$ . In this paper, we mainly investigate the type-1 GV tensors including their products, their spectra, and their positivities. Applications of GV tensors are also introduced.

Stimulated by odd-bipartite and even-bipartite hypergraphs, we define odd-bipartite (weakly odd-bipartie) and even-bipartite (weakly evenbipartite) 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 Z-tensor with nonnegative diagonal elements, and the largest H-eigenvalue of absolute tensor of that Ztensor. When the order is even and the 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 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 odd-bipartite. 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.

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 first study properties of lk,s-singular values of real rectangular tensors. Then, a necessary and sufficient condition for the positive definiteness of partially symmetric rectangular tensors is given. Furthermore, we show that the weak Perron-Frobenius theorem for nonnegative partially symmetric rectangular tensor keeps valid under some new conditions and we prove a maximum property for the largest lk,s-singular values of nonnegative partially symmetric rectangular tensor. Finally, we prove that the largest lk,s-singular value of nonnegative weakly irreducible partially symmetric rectangular tensor is still geometrically simple.

The H-matrices are an important class in the matrix theory, and have many applications. Recently, this concept has been extended to higher order ℋ-tensors. In this paper, we establish important properties of diagonally dominant tensors and ℋ-tensors. Distributions of eigenvalues of nonsingular symmetric ℋ-tensors are given. An ℋ+-tensor is semi-positive, which enlarges the area of semi-positive tensor from M-tensor to ℋ+-tensor. The spectral radius of Jacobi tensor of a nonsingular (resp. singular) ℋ-tensor is less than (resp. equal to) one. In particular, we show that a quasi-diagonally dominant tensor is a nonsingular ℋ-tensor if and only if all of its principal sub-tensors are nonsingular ℋ-tensors. An irreducible tensor A is an ℋ-tensor if and only if it is quasi-diagonally dominant.

We investigate k-uniform loose paths. We show that the largest H-eigenvalues of their adjacency tensors, Laplacian tensors, and signless Laplacian tensors are computable. For a k-uniform loose path with length l ⩾ 3, we show that the largest H-eigenvalue of its adjacency tensor is \({\left( {{\raise0.7ex\hbox{${\left( {1 + \sqrt 5 } \right)}$} \!\mathord{\left/ {\vphantom {{\left( {1 + \sqrt 5 } \right)} 2}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$2$}}} \right)^{{\raise0.7ex\hbox{$2$} \!\mathord{\left/ {\vphantom {2 k}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$k$}}}}\) when l = 3 and λ(A) = 31/k when l = 4, respectively. For the case of l ⩾ 5, we tighten the existing upper bound 2. We also show that the largest H-eigenvalue of its signless Laplacian tensor lies in the interval (2, 3) when l ⩾ 5. Finally, we investigate the largest H-eigenvalue of its Laplacian tensor when k is even and we tighten the upper bound 4.

We study iterative methods for solving a set of sparse non-negative tensor equations (multivariate polynomial systems) arising from data mining applications such as information retrieval by query search and community discovery in multi-dimensional networks. By making use of sparse and non-negative tensor structure, we develop Jacobi and Gauss-Seidel methods for solving tensor equations. The multiplication of tensors with vectors are required at each iteration of these iterative methods, the cost per iteration depends on the number of non-zeros in the sparse tensors. We show linear convergence of the Jacobi and Gauss-Seidel methods under suitable conditions, and therefore, the set of sparse non-negative tensor equations can be solved very efficiently. Experimental results on information retrieval by query search and community discovery in multi-dimensional networks are presented to illustrate the application of tensor equations and the effectiveness of the proposed methods.

Based on the structure of the rank-1 matrix and the different unfolding ways of the tensor, we present two types of structured tensors which contain the rank-1 tensors as special cases. We study some properties of the ranks and the best rank-r approximations of the structured tensors. By using the upper-semicontinuity of the matrix rank, we show that for the structured tensors, there always exist the best rank-r approximations. This can help one to better understand the sequential unfolding singular value decomposition (SVD) method for tensors proposed by J. Salmi et al. [IEEE Trans Signal Process, 2009, 57(12): 4719–4733] and offer a generalized way of low rank approximations of tensors. Moreover, we apply the structured tensors to estimate the upper and lower bounds of the best rank-1 approximations of the 3rd-order and 4th-order tensors, and to distinguish the well written and non-well written digits.

International audience; Non-negative tensor decomposition allows us to analyze data in their ‘native’ form and to present results in the form of the sum of rank-1 tensors that does not nullify any parts of the factors. In this paper, we propose the geometrical structure of a basis vector frame for sum-of-rank-1 type decomposition of real-valued non-negative tensors. The decomposition we propose reinterprets the orthogonality property of the singular vectors of matrices as a geometric constraint on the rank-1 matrix bases which leads to a geometrically constrained singular vector frame. Relaxing the orthogonality requirement, we developed a set of structured-bases that can be utilized to decompose any tensor into a similar constrained sum-of-rank-1 decomposition. The proposed approach is essentially a reparametrization and gives us an upper bound of the rank for tensors. At first, we describe the general case of tensor decomposition and then extend it to its non-negative form. At the end of this paper, we show numerical results which conform to the proposed tensor model and utilize it for non-negative data decomposition.

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.

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 Heigenvectors in general. We prove that for any m ⩾ 2, an m-order 2-dimensional tensor A exists such that A 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. Furthermore, 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-eigenvectors 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-eigenvectors.

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.