Bilinear Factor Matrix Norm Minimization for Robust PCA: Algorithms and Applications

Area of Science:

• Computer Vision
• Machine Learning
• Optimization

Background:

• Heavy-tailed distributions of corrupted outliers and singular values are effective priors in low-level vision but lead to computationally challenging optimization problems.
• Existing algorithms struggle with non-convex, non-smooth, and non-Lipschitz problems, limiting their scalability for large-scale applications.

Purpose of the Study:

• To develop more tractable and scalable optimization models for robust principal component analysis (RPCA) using novel matrix norm penalties.
• To enhance the performance of RPCA in low-level vision applications by addressing the limitations of existing methods.

Main Methods:

• Proposed two novel bilinear factor matrix norm minimization models for RPCA.
• Defined double nuclear norm and Frobenius/nuclear hybrid norm penalties.
• Proved these penalties are equivalent to Schatten-1/2 and 2/3 quasi-norms, resulting in Lipschitz optimization problems.

Main Results:

• The proposed methods yield more accurate solutions than original Schatten quasi-norm minimization, even with limited observations.
• The new models lead to more tractable and scalable Lipschitz optimization problems.
• Experimental analysis demonstrates superior performance in various low-level vision tasks.

Conclusions:

• The novel bilinear factor matrix norm minimization models offer significant improvements in accuracy and scalability for RPCA.
• These methods effectively address the challenges posed by heavy-tailed distributions in low-level vision.
• The proposed penalties outperform state-of-the-art methods in applications such as text removal, object detection, image alignment, and inpainting.