Matrix Completion Based on Non-convex Low Rank Approximation.

Area of Science:

• Applied Mathematics
• Computer Science
• Data Science

Background:

• Nuclear Norm Minimization (NNM) is a common technique for matrix completion and low rank approximation.
• NNM's limitations include ignoring singular value differences, leading to suboptimal results.
• Existing methods often lack efficiency and convergence guarantees.

Purpose of the Study:

• To develop a more effective regularizer for matrix completion beyond NNM.
• To construct efficient optimization models for solving the proposed regularizer.
• To demonstrate the applicability of the new method to related problems like subspace clustering.

Main Methods:

• Introduction of a novel non-convex regularizer for Rank Minimization (RM).
• Development of two matrix completion models utilizing the new regularizer.
• Design of an efficient optimization algorithm with convergence guarantees.
• Application of the regularizer and method to subspace clustering.

Main Results:

• The proposed non-convex regularizer outperforms NNM in matrix completion tasks.
• The developed optimization method achieves faster convergence than conventional approaches.
• Experimental results on real images show significant advantages over state-of-the-art algorithms.
• The method demonstrates effectiveness in subspace clustering based on low rank representation.

Conclusions:

• The novel non-convex regularizer and associated optimization method offer significant improvements for matrix completion.
• The approach is versatile and applicable to other Rank Minimization problems.
• The developed method provides a faster and more accurate solution compared to existing techniques.