Fast Robust Matrix Completion via Entry-Wise ℓ0-Norm Minimization

• Published in: IEEE transactions on cybernetics Vol. 53; no. 11; pp. 7199 - 7212
• Main Authors: Xiao Peng Li, Zhang-Lei, Shi, Liu, Qi, Hing Cheung So
• Format: Journal Article
• Language: English
• Published: Piscataway The Institute of Electrical and Electronics Engineers, Inc. (IEEE) 01.11.2023
• Subjects: Algorithms; Anomalies; Computational efficiency; Computing time; Data analysis; Hyperspectral imaging; Optimization; Outliers (statistics); Random noise; Recovery; Robustness
• ISSN: 2168-2267; 2168-2275
• DOI: 10.1109/TCYB.2022.3224070

Matrix completion (MC) aims at recovering missing entries, given an incomplete matrix. Existing algorithms for MC are mainly designed for noiseless or Gaussian noise scenarios and, thus, they are not robust to impulsive noise. For outlier resistance, entry-wise [Formula Omitted]-norm with [Formula Omitted] and M-estimation are two popular approaches. Yet the optimum selection of [Formula Omitted] for the entrywise [Formula Omitted]-norm-based methods is still an open problem. Besides, M-estimation is limited by a breakdown point, that is, the largest proportion of outliers. In this article, we adopt entrywise [Formula Omitted]-norm, namely, the number of nonzero entries in a matrix, to separate anomalies from the observed matrix. Prior to separation, the Laplacian kernel is exploited for outlier detection, which provides a strategy to automatically update the entrywise [Formula Omitted]-norm penalty parameter. The resultant multivariable optimization problem is addressed by block coordinate descent (BCD), yielding [Formula Omitted]-BCD and [Formula Omitted]-BCD-F. The former detects and separates outliers, as well as its convergence is guaranteed. In contrast, the latter attempts to treat outlier-contaminated elements as missing entries, which leads to higher computational efficiency. Making use of majorization–minimization (MM), we further propose [Formula Omitted]-BCD-MM and [Formula Omitted]-BCD-MM-F for robust non-negative MC where the nonnegativity constraint is handled by a closed-form update. Experimental results of image inpainting and hyperspectral image recovery demonstrate that the suggested algorithms outperform several state-of-the-art methods in terms of recovery accuracy and computational efficiency.