Low-Rankness Enhanced Robust Tensor Principal Component Analysis: A Nonlinear Monotonic Function Approach

Low-rankness, recognized as an intrinsic property of high-dimensional data, plays a central role in many data recovery tasks, such as robust tensor principal component analysis (RTPCA). The main idea of RTPCA is to employ an appropriate convex or nonconvex tensor nuclear norm (TNN) to emphasize significant singular values while suppressing redundant components, thereby promoting low-rank structure of the target tensor. However, both existing convex TNNs and nonconvex TNNs still exhibit certain limitations: convex TNNs are very likely to result in suboptimal recovery due to their uniform shrinkage of all singular values, while nonconvex TNNs may retain residual noise due to insufficient suppression of large singular values. To address these issues, a tailored nonlinear monotonic weighting (TNMW) function is developed, based on which a low-rankness-enhanced RTPCA framework is proposed. The TNMW function consists of four key components: a scale-invariant mechanism (SIM) to improve model generalization, a fractional attenuation mechanism (FAM) to initially exploit low-rank structure, a logarithmic enhancement mechanism (LEM) to further promote low-rankness, and an exponential regulation mechanism (ERM) to ensure monotonic controllability. Benefiting from these designs, the TNMW function exhibits desirable theoretical properties, including monotonicity, smoothness, Lipschitz continuity, well-behaved boundary responses, and local convexity under appropriate parameter settings. Based on these properties, a weighted TNN (WTNN) is defined, and the global optimality of the corresponding WTNN minimization problem is established. Finally, the extensive experiments on several public datasets demonstrate the superiority of the proposed RTPCA method compared with several state-of-the-art approaches in image recovery and background modeling.