In this paper, we study an eigenvalue problem for even order tensors. Using the matrix unfolding of even order tensors, we can establish the relationship between a tensor eigenvalue problem and a multilevel matrix eigenvalue problem. By considering a higher order singular value decomposition of a tensor, we show that higher order singular values are the square root of the eigenvalues of the product of the tensor and its conjugate transpose. This result is similar to that in matrix case. Also we study an eigenvalue problem for Toeplitz/circulant tensors, and give the lower and upper bounds of eigenvalues of Toeplitz tensors. An application in image restoration is also discussed.

In this paper, we study the global uniqueness and solvability (GUS-property) of tensor complementarity problems (TCPs) for some special structured tensors. The modulus equation for TCPs is also proposed, and based on this equation, we develop the corresponding nonsmooth Newton’s method, which extends the existing method given in the work of Zheng H and Li W. The modulus-based nonsmooth Newton’s method for solving linear complementarity problems. J Comput Appl Math. 2015;288: 116-126. Numerical examples are given to demonstrate the efficiency of the proposed algorithm.

Various image datasets appear naturally in the form of multi-dimensional arrays (hypermatrices), called tensors. Image with incomplete entries, which often can be formulated as the low-rank tensor completion problem, is practically important in order to process large size 3D array images both efficiently and effectively. To make the low-rank approximation being tractable, most current approaches make use of the lower convex envelope of tensor multi-rank function for the approximation. Due to the gap between the rank function and its lower convex envelope, the adoption of tensor nuclear norm may lead to the approximation of the corresponding tensor tubal-rank being insufficient. In this paper, we introduce a new nonconvex regularization approach, which can better capture the low-rank characteristics than the convex approach. By transforming the original problem to the Fourier domain, we formulate an equivalent optimization problem with more transparent tensor rank characteristics, whose explicit solution can be obtained under our framework. A minimization algorithm, associated with the augmented Lagrangian multipliers and the nonconvex regularizer, is established and is shown to be feasible. The constructed sequence converges to the desirable Karush-Kuhn-Tucker point, which is mathematically validated in detail. Extensive experimental results demonstrate that our proposed approach outperforms the existing state-of-the-art convex approaches consistently.