An Iterative Reweighted Method for Tucker Decomposition of Incomplete Tensors

We consider the problem of low-rank decomposition of incomplete 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 $N$th-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 proposed method is developed by iteratively minimizing a surrogate function that majorizes the original objective function. This iterative optimization leads to an iteratively reweighted least squares algorithm. 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.