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.

我们提出了一种新的对数极小极大函数，该函数可以在保护大奇异值的同时对小奇异值施加更强的惩罚。基于此，我们为张量管秩定义了一种加权张量LM范数作为张量的非凸松弛。然后，分别提出了基于TLM的低秩张量完成模型以及基于TLM的张量鲁棒主成分分析模型。此外，我们对所提出的方法提供了理论收敛性保证。通过对各种真实数据集进行全面实验，并与相似的EMLCP方法进行比较分析，结果表明所提出的方法优于现有方法。

基于对数极小值函数的非凸三阶张量恢复