  We consider the problem of low-rank decomposition of incomplete multiway
tensors. Since many real-world data lie on an intrinsically low dimensional
subspace, tensor low-rank decomposition with missing entries has applications
in many data analysis problems such as recommender systems and image
inpainting. In this paper, we focus on Tucker decomposition which represents an
Nth-order tensor in terms of N factor matrices and a core tensor via
multilinear operations. To exploit the underlying multilinear low-rank
structure in high-dimensional datasets, we propose a group-based log-sum
penalty functional to place structural sparsity over the core tensor, which
leads to a compact representation with smallest core tensor. The method for
Tucker decomposition is developed by iteratively minimizing a surrogate
function that majorizes the original objective function, which results in an
iterative reweighted process. In addition, to reduce the computational
complexity, an over-relaxed monotone fast iterative shrinkage-thresholding
technique is adapted and embedded in the iterative reweighted process. The
proposed method is able to determine the model complexity (i.e. multilinear
rank) in an automatic way. Simulation results show that the proposed algorithm
offers competitive performance compared with other existing algorithms.
