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Matrix completion from very limited information is an important machine learning topic, and has received extensive attention in various scientific applications. Matrix completion aims at finding a low-rank matrix to approximate the incomplete data matrix. However, noise in the data matrix may degrade the performance of the existing matrix completion algorithms, especially if there are different types of noise. In this paper, we proposed a robust matrix completion method with column outliers and sparse noise. The incomplete matrix is iteratively divided into low-rank and sparse parts. The ℓ 2 , 1 -norm based objective function makes the recovered matrix keeps a low-rank structure and lets the algorithm robust to column outliers, while the regularization term based on ℓ 1 -norm can alleviate the influence of sparse noise. Besides, a vector completion algorithm has been proposed to help us estimate the missing entries of the out-of-sample vectors. Moreover, the proposed model can be optimized by an efficient iterative re-weighted method, without introducing any additional parameters, while the adaptive weights obtained in the optimization process can help us detect column outliers. Both theoretical analysis and experiments based on synthetic datasets and real world datasets are implemented to validate the performance of the proposed method. [ABSTRACT FROM AUTHOR]

Fuzzy c-means (FCM) is a classic unsupervised clustering algorithm in machine learning fields. Euclidean distance is a frequently used distance metric in FCM, but it is only suitable for data with spherical clusters. Therefore, Mahalanobis distance was introduced into Gustafson-Kessel Fuzzy C-Means (GK-FCM) to help improve the performance on data with ellipsoidal clusters. However, GK-FCM and existing Mahalanobis distance based algorithms only focus on squared Mahalanobis distance, because squared Mahalanobis distance based problems are usually convex and easily solvable. But squared Mahalanobis distance is not a perfect metric, because it tends to exaggerate the influence of outliers and lead to unsatisfying results. In this paper, we propose a rooted Mahalanobis distance based GK-FCM model, which has better clustering performance and superior robustness than traditional GK-FCM. Moreover, owing to the introduction of rooted Mahalanobis distance, the optimization of the proposed model becomes non-trivial and it is not realistic to obtain a closed-form solution as that of traditional GK-FCM. In this paper, by making reference to the re-weighted method, we develop a novel iterative converging algorithm to optimize the proposed model. Finally, extensive experiments are conducted on both synthetic and real-world data sets to manifest the superiority of the proposed model. [ABSTRACT FROM AUTHOR]