Tensor p-shrinkage nuclear norm for low-rank tensor completion
This work addresses tensor completion problems, which are important for data analysis in fields like computer vision and signal processing, but it is incremental as it builds on existing tensor decomposition frameworks.
The authors tackled low-rank tensor completion by proposing a new tensor p-shrinkage nuclear norm (p-TNN) that better approximates tensor rank than existing norms, leading to a model with a proven recovery error bound and superior performance over state-of-the-art methods in numerical experiments.
In this paper, a new definition of tensor p-shrinkage nuclear norm (p-TNN) is proposed based on tensor singular value decomposition (t-SVD). In particular, it can be proved that p-TNN is a better approximation of the tensor average rank than the tensor nuclear norm when p < 1. Therefore, by employing the p-shrinkage nuclear norm, a novel low-rank tensor completion (LRTC) model is proposed to estimate a tensor from its partial observations. Statistically, the upper bound of recovery error is provided for the LRTC model. Furthermore, an efficient algorithm, accelerated by the adaptive momentum scheme, is developed to solve the resulting nonconvex optimization problem. It can be further guaranteed that the algorithm enjoys a global convergence rate under the smoothness assumption. Numerical experiments conducted on both synthetic and real-world data sets verify our results and demonstrate the superiority of our p-TNN in LRTC problems over several state-of-the-art methods.