Iteratively Capped Reweighting Norm Minimization with Global Convergence Guarantee for Low-Rank Matrix Learning

Area of Science:

• Machine Learning
• Computer Vision
• Optimization

Background:

• Low Rank Matrix Learning (LRML) is widely used in machine learning and computer vision.
• Existing LRML methods often use surrogate functions for rank relaxation, leading to suboptimal solutions by ignoring rank component differences.

Purpose of the Study:

• To propose a novel nonconvex regularizer, capped reweighting norm minimization (CRNM), for improved LRML.
• To address the limitations of existing methods by considering distinct rank component contributions and adaptively truncating singular values.

Main Methods:

• Developed a general LRML model incorporating the CRNM regularizer.
• Derived a closed-form solution for the CRNM regularized least squares subproblem under mild conditions.
• Designed an efficient optimization method with convergence guarantees, utilizing the Kurdyka-Łojasiewicz (KŁ) inequality for convergence analysis.

Main Results:

• The CRNM regularizer effectively considers different contributions of rank components and adaptively truncates singular values.
• The proposed optimization method demonstrates high computational efficiency and convergence guarantees.
• The CRNM-based approach shows significant advantages over state-of-the-art methods in matrix completion and subspace clustering tasks.

Conclusions:

• The proposed CRNM regularizer and optimization method offer a powerful and efficient solution for various low rank matrix learning problems.
• CRNM provides a more nuanced approach to rank minimization, leading to improved performance and overcoming limitations of previous LRML techniques.