Guanghui Cheng

Ping-Ping Wang, Liang Li, Guang-Hui Cheng

Tensor completion is a challenging problem with various applications. Many related models based on the low-rank prior of the tensor have been proposed. However, the low-rank prior may not be enough to recover the original tensor from the observed incomplete tensor. In this paper, we prose a tensor completion method by exploiting both the low-rank and sparse prior of tensor. Specifically, the tensor completion task can be formulated as a low-rank minimization problem with a sparse regularizer. The low-rank property is depicted by the tensor truncated nuclear norm based on tensor singular value decomposition (T-SVD) which is a better approximation of tensor tubal rank than tensor nuclear norm. While the sparse regularizer is imposed by a $\ell_{1}$-norm in a discrete cosine transformation (DCT) domain, which can better employ the local sparse property of completed data. To solve the optimization problem, we employ an alternating direction method of multipliers (ADMM) in which we only need to solve several subproblems which have closed-form solutions. Substantial experiments on real world images and videos show that the proposed method has better performances than the existing state-of-the-art methods.

Yunfeng Cai, Guanghui Cheng, Decai Shi

In this paper, we consider the exact/approximate general joint block diagonalization (GJBD) problem of a matrix set $\{A_i\}_{i=0}^p$ ($p\ge 1$), where a nonsingular matrix $W$ (often referred to as diagonalizer) needs to be found such that the matrices $W^{H}A_iW$'s are all exactly/approximately block diagonal matrices with as many diagonal blocks as possible. We show that the diagonalizer of the exact GJBD problem can be given by $W=[x_1, x_2, \dots, x_n]Π$, where $Π$ is a permutation matrix, $x_i$'s are eigenvectors of the matrix polynomial $P(λ)=\sum_{i=0}^pλ^i A_i$, satisfying that $[x_1, x_2, \dots, x_n]$ is nonsingular, and the geometric multiplicity of each $λ_i$ corresponding with $x_i$ equals one. And the equivalence of all solutions to the exact GJBD problem is established. Moreover, theoretical proof is given to show why the approximate GJBD problem can be solved similarly to the exact GJBD problem. Based on the theoretical results, a three-stage method is proposed and numerical results show the merits of the method.