Robust Low-Rank Tensor Recovery via Nonconvex Singular Value Minimization

Area of Science:

• Tensor analysis
• Machine learning
• Data science

Background:

• Tensor robust principal component analysis (RPCA) using tensor nuclear norm (TNN) minimization is effective for recovering low-rank tensors corrupted by sparse noise.
• However, TNN's convex nature can lead to over-penalization of singular values, resulting in biased solutions.

Purpose of the Study:

• To propose a novel nonconvex surrogate for tensor rank, the tensor logarithmic norm (TLN), to mitigate bias in low-rank tensor recovery.
• To enhance robustness against outliers using a nonconvex 'p-ball projection scheme.
• To develop efficient algorithms for low-rank tensor recovery using TLN minimization and tensor factorization.

Main Methods:

• Introduced a new tensor logarithmic norm (TLN) as a nonconvex surrogate for tensor rank.
• Incorporated tensor factorization into TLN minimization for improved computational efficiency.
• Developed a nonconvex 'p-ball projection scheme (0 < p < 1) for enhanced outlier robustness.
• Utilized the alternating direction method of multipliers (ADMM) to solve the resulting optimization problems, providing convergence guarantees.

Main Results:

• The proposed TLN-based methods effectively reduce bias associated with large singular values compared to TNN.
• The nonconvex 'p-ball projection significantly improves robustness in impulsive scenarios with outliers.
• Experimental results on synthetic data and real-world image/video restoration demonstrate superior performance over state-of-the-art algorithms.
• Achieved higher tensor recovery accuracy and computational efficiency.

Conclusions:

• The proposed TLN minimization combined with tensor factorization and nonconvex projection offers a powerful approach for robust low-rank tensor recovery.
• These novel methods address limitations of existing TNN-based techniques, particularly regarding bias and outlier handling.
• The developed algorithms show significant promise for applications in data science, machine learning, and signal processing requiring accurate tensor decomposition.