A Fast and Accurate Matrix Completion Method Based on QR Decomposition and L 2,1 -Norm Minimization

• Published in: IEEE transaction on neural networks and learning systems Vol. 30; no. 3; pp. 803 - 817
• Main Authors: Liu, Qing, Davoine, Franck, Yang, Jian, Cui, Ying, Jin, Zhong, Han, Fei
• Format: Journal Article
• Language: English
• Published: United States IEEE 01.03.2019
• Subjects: Computer Science; Computer Vision and Pattern Recognition; Image Processing; QR Decomposition; Iteratively Reweighted L21-Norm; Matrix Completion; Qatar Riyal (QR) decomposition
• ISSN: 2162-237X; 2162-2388
• DOI: 10.1109/TNNLS.2018.2851957

Low-rank matrix completion aims to recover matrices with missing entries and has attracted considerable attention from machine learning researchers. Most of the existing methods, such as weighted nuclear-norm-minimization-based methods and Qatar Riyal (QR)-decomposition-based methods, cannot provide both convergence accuracy and convergence speed. To investigate a fast and accurate completion method, an iterative QR-decomposition-based method is proposed for computing an approximate singular value decomposition. This method can compute the largest singular values of a matrix by iterative QR decomposition. Then, under the framework of matrix trifactorization, a method for computing an approximate SVD based on QR decomposition (CSVD-QR)-based L -norm minimization method (LNM-QR) is proposed for fast matrix completion. Theoretical analysis shows that this QR-decomposition-based method can obtain the same optimal solution as a nuclear norm minimization method, i.e., the L -norm of a submatrix can converge to its nuclear norm. Consequently, an LNM-QR-based iteratively reweighted L -norm minimization method (IRLNM-QR) is proposed to improve the accuracy of LNM-QR. Theoretical analysis shows that IRLNM-QR is as accurate as an iteratively reweighted nuclear norm minimization method, which is much more accurate than the traditional QR-decomposition-based matrix completion methods. Experimental results obtained on both synthetic and real-world visual data sets show that our methods are much faster and more accurate than the state-of-the-art methods.