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The tensor equation A x m - 1 = b with the tensor A of order m and dimension n and the vector b , has practical applications in several fields including signal processing, high-dimensional PDEs, high-order statistics, and so on. In this paper, a class of exponential accelerated iterative methods is proposed for solving the tensor equation mentioned above in the sense that the coefficient tensor A is a symmetric and nonsingular or singular M -tensor. The obtained iterative schemes involve the classical Newton's method as a special case. It is shown that the proposed method for nonsingular case is superlinearly convergent, while for singular cases, it is linearly convergent. The performed numerical experiments demonstrate that our methods outperform some existing ones. [ABSTRACT FROM AUTHOR]

In this paper, we study the problem of a batch of linearly correlated image alignment, where the observed images are deformed by some unknown domain transformations, and corrupted by additive Gaussian noise and sparse noise simultaneously. By stacking these images as the frontal slices of a third-order tensor, we propose to utilize the tensor factorization method via transformed tensor-tensor product to explore the low-rankness of the underlying tensor, which is factorized into the product of two smaller tensors via transformed tensor-tensor product under any unitary transformation. The main advantage of transformed tensor-tensor product is that its computational complexity is lower compared with the existing literature based on transformed tensor nuclear norm. Moreover, the tensor ℓ p (0 < p < 1) norm is employed to characterize the sparsity of sparse noise and the tensor Frobenius norm is adopted to model additive Gaussian noise. A generalized Gauss-Newton algorithm is designed to solve the resulting model by linearizing the domain transformations, and a proximal Gauss-Seidel algorithm is developed to solve the corresponding subproblem. Furthermore, the convergence of the proximal Gauss-Seidel algorithm is established according to Kurdyka-Łojasiewicz property, whose convergence rate is also analyzed. Extensive numerical examples on real-world image datasets are carried out to demonstrate the superior performance of the proposed method as compared to several state-of-the-art methods in both accuracy and computational time. [ABSTRACT FROM AUTHOR]

In this paper, we mainly develop the well-known vector and matrix polynomial extrapolation methods in tensor framework. To this end, some new products between tensors are defined and the concept of positive definitiveness is extended for tensors corresponding to T-product. Furthermore, we discuss on the solution of least-squares problem associated with a tensor equation using Tensor Singular Value Decomposition (TSVD). Motivated by the effectiveness of some proposed vector extrapolation methods in earlier papers, we describe how an extrapolation technique can be also implemented on the sequence of tensors produced by truncated TSVD (TTSVD) for solving possibly ill-posed tensor equations. [ABSTRACT FROM AUTHOR]

Multilinear PageRank is a variant of the PageRank algorithm that takes into account multiple relationships among nodes in a network. This algorithm can make web page ranking more efficient and accurate by considering multiple types of connections at once. The higher-order power method is commonly used to calculate the multilinear PageRank vector due to its ease of implementation and low storage needs, and because it is a natural extension of the traditional power method used in the PageRank algorithm. However, the convergence of this method is not guaranteed, and even when it occurs, the process is often slow. In this paper, we show how some vector extrapolation methods such as minimal polynomial extrapolation (MPE) and reduced rank extrapolation (RRE) could be used for accelerating the computation of the fixed-point multilinear PageRank. [ABSTRACT FROM AUTHOR]

In this paper, we consider the generalized nonnegative tensor factorization (GNTF) problem, which arises in multiple-tissue gene expression and multi-target tracking. Based on the Karhsh-Kuhn-Tucker conditions, the necessary condition of the local solution for the GNTF problem is given. The proximal alternating nonnegative least squares method is designed to solve it, and its convergence theorem is also derived. Numerical examples show that the new method is feasible and effective. [ABSTRACT FROM AUTHOR]

Inthis paper, we study the global uniqueness and solvability of tensor complementarity problems for ℋ + -tensors. We obtain a sufficient condition of the global uniqueness and solvability of tensor complementarity problems for ℋ + -tensors. We present nonlinear dynamical system models for solving the tensor complementarity problem (TCP). We prove that the presented dynamical system models are stable in the sense of Lyapunov stability theory for considering three classes of structured tensors. The computer simulation results further substantiate that the considered dynamical system can be used to solve the TCP. [ABSTRACT FROM AUTHOR]

The strong ℋ -tensors have important applications in many areas of science and engineering, e.g., the determination of positive definiteness for an even-order homogeneous polynomial form in the real field. In this paper, we propose two iterative algorithms with non-parameter for identifying strong ℋ -tensors, which overcome the drawback of choosing the best value of parameter 𝜖 in some existing algorithms given by Li et al. and Liu et al. (J. Comput. Appl. Math., 255, 1–14, 2014 and Comput. Appl. Math. 36, 1623–1635, 2017). Some numerical experiments are performed to illustrate the feasibility and effectiveness of our algorithms. [ABSTRACT FROM AUTHOR]

Recently, the alternating direction method of multipliers (ADMM) and its variations have gained great popularity in large-scale optimization problems. This paper is concerned with the solution of the tensor equation Axm−1=b in which A is an m th-order and n-dimensional real tensor and b is an n-dimensional real vector. By introducing certain auxiliary variables, we transform equivalently this tensor equation into a consensus constrained optimization problem, and then propose an ADMM type method for it. It turns out that each limit point of the sequences generated by this method satisfies the Karush-Kuhn-Tucker conditions. Moreover, from the perspective of computational complexity, the proposed method may suffer from the curse-of-dimensionality if the size of the tensor equation is large, and thus we further present a modified version (as a variant of the former) turning to the tensor-train decomposition of the tensor A, which is free from the curse. As applications, we establish the associated inverse iteration methods for solving tensor eigenvalue problems. The performed numerical examples illustrate that our methods are feasible and efficient. [ABSTRACT FROM AUTHOR]

Linear matrix equations such as the Sylvester equation, Lyapunov equation, Stein equation, and a variate of their generalizations are of significant importance in many applications. A conversion to a classical linear system $\mathcal {A}\mathbf {x}=\mathbf {b}$ via the Kronecker product is generally regarded as the last resort because it significantly increases the size of the problem and disrespects any underlying structure. Convention Krylov subspace methods such as GMRES or CGNR might not need the vectorization explicitly, but an otherwise well established preconditioner for $\mathcal {A}$ encounters the difficulty that it must be disassembled and redistributed over the original matrix coefficients in order to complete evade the Kronecker vectorization. Thus many other techniques for solving linear matrix equations have been developed, which are usually problem dependent and can hardly be generalized when the equation is changed. In contrast, motivated by the notion of order-4 tensor equations, this paper proposes the idea of casting any linear matrix equation under the same framework of generalized normal equation and using low-precision gradient dynamics to achieve high-precision solution. A single computational paradigm therefore serves to handle all types of linear matrix equations. The flow approach has the advantages of being straightforward for implementation, uniform in theory, versatile in application, working directly with the original sizes without Kronecker vectorization, avoiding inversion or factorization, and being easy for convergence analysis. This paper outlines the theory, exemplifies a collection of applications, suggests a simple implementation, and reports some numerical evidences. [ABSTRACT FROM AUTHOR]