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The Cauchy relations distinguish between rari- and multi-constant linear elasticity theories. These relations are treated in this paper in a form that is invariant under two groups of transformations: indices permutation and general linear transformations of the basis. The irreducible decomposition induced by the permutation group is outlined. The Cauchy relations are then formulated as a requirement of nullification of an invariant subspace. A successive decomposition under rotation group allows to define the partial Cauchy relations and two types of elastic materials. We explore several applications of the full and partial Cauchy relations in physics of materials. The structure's deviation from the basic physical assumptions of Cauchy's model is defined in an invariant form. The Cauchy and non-Cauchy contributions to Hooke's law and elasticity energy are explained. We identify wave velocities and polarization vectors that are independent of the non-Cauchy part for acoustic wave propagation. Several bounds are derived for the elasticity invariant parameters. [ABSTRACT FROM AUTHOR]

This paper is concerned with the iterative solutions to a class of generalized Sylvester quaternion tensor equations. From the Kronecker product and the properties of quaternion tensors, we propose the biconjugate A-orthogonal residual stabilized and the generalized product-type biconjugate A-orthogonal residual methods in their tensor forms for solving the original tensor equation. As an application, the proposed methods are tested with some colour video restoration problems. Numerical results show the effectiveness of our methods compared with some existing methods. [ABSTRACT FROM AUTHOR]

In this paper, we study that the algebraic multiplicity of the zero Laplacian eigenvalue of a connected uniform hypergraph. We give the algebraic multiplicity of the zero Laplacian eigenvalue of a hyperstar. For a loose hyperpath, we characterize the algebraic multiplicity of the zero Laplacian eigenvalue by the multiplicities of points in the affine variety defined by the Laplacian eigenvalue equations. We compute the algebraic multiplicities of the zero Laplacian eigenvalue of a loose hyperpath. We also show that the algebraic multiplicity of the zero Laplacian eigenvalue is not smaller than the number of irreducible components of the eigenvariety associated with the zero Laplacian eigenvalue for a loose hyperpath. [ABSTRACT FROM AUTHOR]

In magnetic resonance imaging, high angular resolution diffusion imaging (abbr. HARDI) is used to characterize non-Gaussian diffusion processes. One approach to analyzing HARDI data is to model the apparent diffusion coefficient with higher order diffusion tensors from a diffusivity function. An intrinsic property of the diffusivity function is positive semi-definite, which reflects the phenomenon of water molecular diffusion in complicated biological tissue environments. In this paper, we provide a workable criterion for judging the positive semi-definiteness of a diffusivity function and shows that it is effective via two numerical examples. [ABSTRACT FROM AUTHOR]

Recently, solving tensor equations (or multilinear systems) has attracted a lot of attention. This paper investigates the tensor form of the Bi-CGSTAB and Bi-CRSTAB methods, by employing Kronecker product, vectorization, and bilinear operator, to solve the generalized coupled Sylvester tensor equations ∑ i = 1 n (X × 1 A i 1 × 2 A i 2 + Y × 1 B i 1 × 2 B i 2) = E 1 , ∑ i = 1 n (X × 1 C i 1 × 2 C i 2 + Y × 1 D i 1 × 2 D i 2) = E 2 , with no matricization. Also some properties of the new methods are presented. By applying multilinear operator, the proposed methods are extended to the general form. Some numerical examples are provided to compare the efficiency of the investigated methods with some existing popular algorithms. Finally, some concluding remarks are given. [ABSTRACT FROM AUTHOR]

In this paper, we propose several improved uniqueness conditions of solution for multilinear PageRank with order-3 irreducible stochastic tensors. By considering upper and lower bounds of solutions, the refined uniqueness conditions are more effective than the existing ones. Furthermore, as an application, sharp perturbation bounds for multilinear PageRank are also developed basing on the new uniqueness conditions. Numerical examples are given to verify the efficiency of the proposed uniqueness conditions. [ABSTRACT FROM AUTHOR]

Efficient and fast computation of a tensor singular value decomposition (t-SVD) with a few passes over the underlying data tensor is crucial because of its many potential applications. The current/existing subspace randomized algorithms need (2 q + 2) passes over the data tensor to compute a t-SVD, where q is a non-negative integer number (power iteration parameter). In this paper, we propose an efficient and flexible randomized algorithm that can handle any number of passes q, which not necessary need be even. The flexibility of the proposed algorithm in using fewer passes naturally leads to lower computational and communication costs. This advantage makes it particularly appropriate when our task calls for several tensor decompositions or when the data tensors are huge. The proposed algorithm is a generalization of the methods developed for matrices to tensors. The expected/average error bound of the proposed algorithm is derived. Extensive numerical experiments on random and real-world data sets are conducted, and the proposed algorithm is compared with some baseline algorithms. The extensive computer simulation experiments demonstrate that the proposed algorithm is practical, efficient, and in general outperforms the state of the arts algorithms. We also demonstrate how to use the proposed method to develop a fast algorithm for the tensor completion problem. [ABSTRACT FROM AUTHOR]

The strong ellipticity condition (abbr. SE-condition) of the displacement equations of equilibrium for general nonlinearly elastic materials plays an important role in nonlinear elasticity and materials. Existing literature shows that it can be equivalently transformed into the SE-condition of an elasticity tensor, and that the SE-condition of an elasticity tensor holds if and only if all of its M-eigenvalues are positive. In order to judge the strong ellipticity of an elasticity tensor, we in this paper propose a direct method for calculating all M-eigenvalues of an elasticity tensor and show that the direct method is workable by a numerical example. [ABSTRACT FROM AUTHOR]

The Jacobian scheme of a reduced, singular projective plane curve is the zero-dimensional scheme, whose homogeneous ideal is generated by the partials of its defining polynomial. The degree of such a scheme is called the global Tjurina number and, if the curve is not a set of concurrent lines, some upper and lower bounds depending on the degree of the curve and the minimal degree of a Jacobian syzygy, have been given by A.A. du Plessis and C.T.C. Wall. In this paper, we give a complete geometric characterization of conic-line arrangements, with global Tjurina number attaining the upper bound. Furthermore, we characterize conic-line arrangements attaining the lower bound for the global Tjurina number, among all curves with a linear Jacobian syzygy. As an application, we characterize conic-line arrangements with Jacobian scheme equal to an eigenscheme of some ternary tensor, and we study the geometry of their polar maps. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of tensor least squares problem with an invertible linear transform, which arises in image restoration. Based on the operator-bidiagonal procedure, two Paige’s algorithms are designed to solve it. The convergence theorems of the new methods are derived. Numerical experiments are performed to illustrate the feasibility and efficiency of the new methods, including when the algorithm is tested with the synthetic data and on some image restoration problems. Comparisons with some previous methods are also given. [ABSTRACT FROM AUTHOR]

This paper introduces a novel approach to computing logic programming semantics. First, a propositional Herbrand base is represented in a vector space and if-then rules in a program are encoded in a matrix. Then the least fixpoint of a definite logic program is computed by matrix-vector products with a non-linear operation. Second, disjunctive logic programs are represented in third-order tensors and their minimal models are computed by algebraic manipulation of tensors. Third, normal logic programs are represented by matrices and third-order tensors, and their stable models are computed. The result of this paper exploits a new connection between linear algebraic computation and symbolic computation, which has the potential to realize logical inference in huge scale of knowledge bases. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to develop a new algorithm for computing a nonnegative low rank tensor approximation for nonnegative tensors that arise in many multidimensional imaging applications. Nonnegativity is one of the important properties, as each pixel value refers to a nonzero light intensity in image data acquisitions. Our approach is different from classical nonnegative tensor factorization (NTF), which requires each factorized matrix, and/or tensor, to be nonnegative. In this paper, we determine a nonnegative low Tucker rank tensor to approximate a given nonnegative tensor. We propose an alternating projections algorithm for computing such a nonnegative low rank tensor approximation, which is referred to as NLRT. The convergence of the proposed manifold projection method is established. The experimental results for synthetic data and multidimensional images are presented to demonstrate that the performance of NLRT is better than the state-of-the-art NTF methods. [ABSTRACT FROM AUTHOR]

Theory of numerical range and numerical radius for tensors is not studied much in the literature. Ke et al. [Linear Algebra Appl. 508 (2016), 100-132: MR3542984] introduced first the notion of the numerical range of a tensor via the k-mode product. However, the convexity of the numerical range via the k-mode product was not proved by them. In this paper, the notion of numerical range and numerical radius for even-order square tensors via the Einstein product are introduced first. Using the notion of the numerical radius of a tensor, we provide some sufficient conditions for a tensor to be unitary. The convexity of the numerical range is also proved. We also provide an algorithm to plot the numerical range of a tensor. Furthermore, some properties of the numerical range for the Moore–Penrose inverse of a tensor are discussed. [ABSTRACT FROM AUTHOR]

This paper investigates the Moore–Penrose inverse of quaternion tensors via the QT-product. Firstly, the definition of the Moore–Penrose inverse of quaternion tensors under the QT-product is presented. The expression of Moore–Penrose inverse of quaternion tensors via the QT-product is proposed based on the singular value decomposition and illustrated by an example. Finally, the perturbation analysis of the Moore–Penrose inverse of quaternion tensors via the QT-product are described under the conditions of one-sided, two-sided and rank-preserving. Moreover, examples of perturbation bounds in each case are given. [ABSTRACT FROM AUTHOR]

In the paper 'On the Dirac–Frenkel Variational Principle on Tensor Banach Spaces', we provided a geometrical description of manifolds of tensors in Tucker format with fixed multilinear (or Tucker) rank in tensor Banach spaces, that allowed to extend the Dirac–Frenkel variational principle in the framework of topological tensor spaces. The purpose of this note is to extend these results to more general tensor formats. More precisely, we provide a new geometrical description of manifolds of tensors in tree-based (or hierarchical) format, also known as tree tensor networks, which are intersections of manifolds of tensors in Tucker format associated with different partitions of the set of dimensions. The proposed geometrical description of tensors in tree-based format is compatible with the one of manifolds of tensors in Tucker format. [ABSTRACT FROM AUTHOR]

The smallest M-eigenvalue τ M (A) of a fourth-order partial symmetric tensor A plays an important role in judging the strong ellipticity condition (abbr. SE-condition) in elastic mechanics. Specifically, if τ M (A) > 0 , then the SE-condition of A holds. In this paper, we establish lower and upper bounds of τ M (A) via extreme eigenvalues of symmetric matrices and tensors constructed by the entries of A . In addition, when A is an elasticity Z-tensor, we establish lower bounds for τ M (A) via the extreme C-eigenvalues of piezoelectric-type tensors. Finally, numerical examples show the efficiency of our proposed bounds in judging the SE-condition of A . [ABSTRACT FROM AUTHOR]

Sparse representation has garnered significant attention across multiple fields, including signal processing, statistics, and machine learning. The fundamental concept of this technique is that we can express the signal as a linear combination of only a few elements from a known basis. Compressed sensing (CS) is an interesting application of this technique. It is valued for its potential to improve data collection and ensure efficient acquisition and recovery from just a few measurements. In this paper, we propose a novel approach for the high-order CS problem based on the Einstein product, utilizing a tensor dictionary instead of the commonly used matrix-based dictionaries in the Tucker model. Our approach provides a more general framework for compressed sensing. We present two novel models to address the CS problem in the multidimensional case. The first model represents a natural generalization of CS to higher-dimensional signals; we extend the traditional CS framework to effectively capture the sparsity of multidimensional signals and enable efficient recovery. In the second model, we introduce a complexity reduction technique by utilizing a low-rank representation of the signal. We extend the OMP and the homotopy algorithms to solve the high-order CS problem. Through various simulations, we validate the effectiveness of our proposed method, including its application to solving the completion tensor problem in 2D and 3D colored and hyperspectral images. [ABSTRACT FROM AUTHOR]

In this paper, we establish some criteria for strong -tensors. The results obtained in this paper extend the corresponding conclusions for strong -matrices and improve the existing results for strong -tensors. As an application, some sufficient conditions of the positive definiteness for an even-order real symmetric tensor are presented. Some numerical results show the feasibility and effectiveness of the results which are obtained in this paper. [ABSTRACT FROM AUTHOR]

In this paper, we obtain some sharp upper and lower bounds for the spectral radii of nonnegative k-uniform tensors (resp., nonnegative tensors) by using the i-th q-times-average slice sum which was introduced by C. Lü, L. H. You and Y. F. Huang [AIMS Math., 2020, 5(3): 1799–1819]. When k = 2, q =1, our results can deduce the main results of D. P. Huang and L. H. You [J. Appl. Math., 2016, 3: 1–7]. We also find that the upper bounds of nonnegative k-uniform tensors we obtained can not be compared with the upper bounds presented by [AIMS Math., 2020, 5(3): 1799–1819] by numerical examples. As applications, we obtain some sharp upper bounds for the α-spectral radii of k-uniform hypergraphs (resp., k-uniform directed hypergraphs). [ABSTRACT FROM AUTHOR]

In the present paper we compare two different iterative approaches to constructing nonnegative tensor train and Tucker decompositions. The first approach is based on idea of alternating projections and randomized sketching for factorization of tensors with nonnegative elements. This approach can be useful for both TT and Tucker formats. The second approach consists of two stages. At the first stage we find the unconstrained tensor train decomposition for the target array. At the second stage we use this initial approximation in order to fix it within moderate number of operations and obtain the factorization with nonnegative factors either in tensor train or Tucker model. We study the performance of these methods for both synthetic data and hyper-spectral image and demonstrate the clear advantage of the latter technique in terms of computational time and wider range of possible applications. [ABSTRACT FROM AUTHOR]

In this paper, a method for quartic minimization over the sphere is studied. It is based on an equivalent difference of convex (DC) reformulation of this problem in the matrix variable. This derivation also induces a global optimality certification for the quartic minimization over the sphere. An algorithm with the subproblem being solved by a semismooth Newton method is then proposed for solving the quartic minimization problem. The efficiency of this algorithm and the global optimality certification are illustrated by numerical experiments. [ABSTRACT FROM AUTHOR]

A d-dimensional tensor A of format n × n × ⋯ × n defines naturally a rational map Ψ from the projective space P n − 1 to itself and its eigenscheme is then the subscheme of P n − 1 of fixed points of Ψ. The eigendiscriminant is an irreducible polynomial in the coefficients of A that vanishes for a given tensor if and only if its eigenscheme is singular. In this paper, we contribute two formulas for the computation of eigendiscriminants in the cases n = 3 and n = 4. In particular, by restriction to symmetric tensors, we obtain closed formulas for the eigendiscriminants of plane curves and surfaces in P 3 as the ratio of some determinants of resultant matrices. [ABSTRACT FROM AUTHOR]

In this paper, we transform the classical linear discriminant analysis (LDA) into a smooth difference-of-convex optimization problem. Then, a new difference-of-convex algorithm with extrapolation is introduced and the convergence of the algorithm is established. Finally, for a face recognition problem, the proposed algorithm achieves better classification performance compared with several current algorithms in the literature. [ABSTRACT FROM AUTHOR]

Tensor completion aims to reconstruct a high-dimensional data from the partial element missing tensors under a low-rank constraint, which may be seen as a least-squares problem on manifold. In this paper, we propose a new Riemannian conjugate gradient method for the tensor completion which performs Riemannian optimization techniques on a fixed transformed multi-rank tensor manifold. More specifically, we generalize the classical Dai-Yuan conjugate gradient method from the Euclidean space to the manifold with fixed rank, and developed within the framework of retraction-based optimization on manifold. Finally, the convergence properties of the proposed algorithm are derived. Numerical experiments with synthetic data and real images demonstrate the feasibility and effectiveness of our approach. [ABSTRACT FROM AUTHOR]