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

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 ℓ_p-norm with 0 < p < 2 and M-estimation are two popular approaches. Yet the optimum selection of p for the entrywise ℓ_p-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 ℓ₀-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 ℓ₀ -norm penalty parameter. The resultant multivariable optimization problem is addressed by block coordinate descent (BCD), yielding ℓ₀-BCD and ℓ₀-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 ℓ₀-BCD-MM and ℓ₀-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.