A Fast and Accurate Matrix Completion Method Based on QR Decomposition and L_2,1 -Norm Minimization

Area of Science:

• Machine Learning
• Numerical Analysis
• Data Science

Background:

• Low-rank matrix completion is crucial for recovering missing data in various applications.
• Existing methods like nuclear norm minimization and QR-decomposition often lack optimal convergence speed and accuracy.
• There is a need for efficient and precise matrix completion techniques.

Purpose of the Study:

• To develop a fast and accurate matrix completion method using iterative QR decomposition.
• To compute an approximate singular value decomposition (SVD) for improved matrix recovery.
• To enhance existing methods for better performance in handling incomplete matrices.

Main Methods:

• An iterative QR-decomposition-based method is proposed for approximate SVD computation.
• A matrix trifactorization framework is used to develop a QR decomposition-based L2,1-norm minimization method (LNM-QR) for fast matrix completion.
• An iteratively reweighted L2,1-norm minimization method (IRLNM-QR) is introduced to boost LNM-QR's accuracy.

Main Results:

• Theoretical analysis confirms that the proposed QR-decomposition method achieves the same optimal solution as nuclear norm minimization.
• IRLNM-QR demonstrates comparable accuracy to iteratively reweighted nuclear norm minimization, significantly outperforming traditional QR-decomposition methods.
• Experiments on synthetic and real-world data show superior speed and accuracy compared to state-of-the-art methods.

Conclusions:

• The proposed LNM-QR and IRLNM-QR methods offer significant improvements in both speed and accuracy for low-rank matrix completion.
• These novel QR-decomposition-based approaches provide a viable alternative to existing matrix completion techniques.
• The findings have implications for applications requiring efficient and precise matrix recovery from incomplete datasets.