Nonconvex third-order Tensor Recovery Based on Logarithmic Minimax Function

Recent researches have shown that low-rank tensor recovery based non-convex relaxation has gained extensive attention. In this context, we propose a new Logarithmic Minimax (LM) function. The comparative analysis between the LM function and the Logarithmic, Minimax concave penalty (MCP), and Minimax Logarithmic concave penalty (MLCP) functions reveals that the proposed function can protect large singular values while imposing stronger penalization on small singular values. Based on this, we define a weighted tensor LM norm as a non-convex relaxation for tensor tubal rank. Subsequently, we propose the TLM-based low-rank tensor completion (LRTC) model and the TLM-based tensor robust principal component analysis (TRPCA) model respectively. Furthermore, we provide theoretical convergence guarantees for the proposed methods. Comprehensive experiments were conducted on various real datasets, and a comparison analysis was made with the similar EMLCP method. The results demonstrate that the proposed method outperforms the state-of-the-art methods.