A novel non-convex minimax $p$-th order concave penalty function approach to low-rank tensor completion

The low-rank tensor completion (LRTC) problem aims to reconstruct a tensor from partial sample information, which has attracted significant interest in a wide range of practical applications such as image processing and computer vision. Among the various techniques employed for the LRTC problem, non-convex relaxation methods have been widely studied for their effectiveness in handling tensor singular values, which are crucial for accurate tensor recovery. While the minimax concave penalty (MCP) non-convex relaxation method has achieved promising results in tackling the LRTC problem and gained widely adopted, it exhibits a notable limitation: insufficient penalty on small singular values during the singular value handling process, resulting in inefficient tensor recovery. To address this issue and enhance recovery performance, a novel minimax $p$-th order concave penalty (MPCP) function is proposed. Based on this novel function, a tensor $p$-th order $\tau$ norm is proposed as a non-convex relaxation for tensor rank approximation, thereby establishing an MPCP-based LRTC model. Furthermore, theoretical convergence guarantees are rigorously established for the proposed method. Extensive numerical experiments conducted on multiple real datasets demonstrate that the proposed method outperforms the state-of-the-art methods in both visual quality and quantitative metrics.