A novel nonconvex approach to recover the low-tubal-rank tensor data: when t-SVD meets PSSV

A recently developed novel tensor decomposition scheme named tensor singular value decomposition (t-SVD) results in a notion of rank referred to as the tubal-rank. Many methods minimize its convex surrogate the tensor nuclear norm (TNN) to enhance the low tubal-rankness of the underlying data. Generally, minimizing the TNN may cause some biases. In this paper, to alleviate these bias phenomenons, we consider to minimize the proposed partial sum of the tensor nuclear norm (PSTNN) in place of the TNN. The novel PSTNN is applied to the tasks of tensor completion and tensor principal component analysis. Numerical experiments are conducted on the synthetic data and real-world data, and experimental results reveal the effectiveness of the proposed methods.