Low-rank tensor completion via a novel minimax $p$-th order concave penalty function

Low-rank tensor completion (LRTC) has attracted significant attention in fields such as computer vision and pattern recognition. Among the various techniques employed in LRTC, non-convex relaxation methods have been widely studied for their effectiveness in handling tensor singular values, which are crucial for accurate tensor recovery. However, the minimax concave penalty (MCP) function, a commonly used non-convex relaxation, exhibits a critical limitation: it effectively preserves large singular values but inadequately processes small ones. To address this issue, a novel minimax $p$-th order concave penalty (MPCP) function is proposed. Building on this advancement, a tensor $p$-th order $\tau$ norm is proposed as a non-convex relaxation for tensor rank estimation, thereby establishing an MPCP-based LRTC model. Furthermore, theoretical guarantees of convergence are provided for the proposed method. Experimental results on multiple real datasets demonstrate that the proposed method outperforms the state-of-the-art methods in both visual quality and quantitative metrics.