Low-Rank Matrix Recovery Via Robust Outlier Estimation

In practice, high-dimensional data are typically sampled from low-dimensional subspaces, but with an intrusion of outliers and/or noises. Recovering the underlying structure and the pollution from the observations is of utmost importance to understanding the data. Besides properly modeling the subspace structure, how to handle the pollution is a core question regarding the recovery quality, the main origins of which include small dense noises and gross sparse outliers. Compared with the small noises, the outliers more likely ruin the recovery, as their arbitrary magnitudes can dominate the fidelity, and thus lead to misleading/erroneous results. Concerning the above, this paper concentrates on robust outlier estimate for low-rank matrix recovery, termed ROUTE. The principle is to classify each entry as an outlier or an inlier (with confidence). We formulate the outlier screening and the recovery into a unified framework. To seek the optimal solution to the problem, we first introduce a block coordinate descent-based optimizer (ROUTE-BCD) and then customize an alternating direction method of multipliers based one (ROUTE-ADMM). Through analyzing theoretical properties and practical behaviors, ROUTE-ADMM shows its superiority over ROUTE-BCD in terms of computational complexity, initialization insensitivity, and recovery accuracy. Extensive experiments on both synthetic and real data are conducted to show the efficacy of our strategy and reveal its significant improvement over other state-of-the-art alternatives.