提出新型非凸惩罚函数，提升低秩张量补全的恢复精度。
A novel non-convex minimax $p$-th order concave penalty function approach to low-rank tensor completion
- 设计$p$阶凹惩罚函数，更好处理小奇异值
- 在真实数据集上显著优于现有最优方法
- 理论保证收敛性，适合图像与视觉任务
低秩张量补全（LRTC）旨在从部分观测数据中重构完整张量，在图像处理与计算机视觉等领域具有广泛应用。现有非凸松弛方法虽有效，但对小奇异值惩罚不足，影响恢复效果。本文提出一种新型极小极大$p$阶凹惩罚（MPCP）函数，构建基于MPCP的张量$p$阶$τ$范数作为非凸秩逼近，并建立相应的LRTC模型。理论分析证明了算法的收敛性。在多个真实数据集上的大量实验表明，该方法在视觉质量和定量指标上均优于当前最优方法。
原文摘要 · Abstract (English)
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 $τ$ 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.
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