Robust Matrix Completion With Column Outliers

Matrix completion, in essence, involves recovering a low-rank matrix from a subset of its entries. Most existing methods for matrix completion neglect two significant issues. First, in several practical applications, such as collaborative filtering, some samples may be corrupted completely. However, most of the robust algorithms consider only the condition that a few components of each column have been arbitrarily contaminated. Second, many real data are not static in nature. Nevertheless, the conventional batch-based matrix completion methods cannot efficiently deal with the out-of-sample, that is, the vector completion problem. In this article, we first provide a novel robust matrix completion model and then develop an efficient optimization method that only requires conducting one time singular value decomposition for a thin matrix per iteration. Furthermore, by exploiting the essence of online matrix completion algorithms, we develop a vector completion model which can help users predict the missing values of out of sample. Numerical comparisons with traditional batch-based and online matrix completion algorithms demonstrate the benefits of the proposed method on streaming data corrupted by column outliers. Moreover, we show that our model can be used to detect outliers from incomplete information. This advantage is validated via numerous experimental results on synthetic and real-world data.