Robust Recovery of Corrupted Low-RankMatrix by Implicit Regularizers

Low-rank matrix recovery algorithms aim to recover a corrupted low-rank matrix with sparse errors. However, corrupted errors may not be sparse in real-world problems and the relationship between ℓ
₁
regularizer on noise and robust M-estimators is still unknown. This paper proposes a general robust framework for low-rank matrix recovery via implicit regularizers of robust M-estimators, which are derived from convex conjugacy and can be used to model arbitrarily corrupted errors. Based on the additive form of half-quadratic optimization, proximity operators of implicit regularizers are developed such that both low-rank structure and corrupted errors can be alternately recovered. In particular, the dual relationship between the absolute function in ℓ
₁
regularizer and Huber M-estimator is studied, which establishes a connection between robust low-rank matrix recovery methods and M-estimators based robust principal component analysis methods. Extensive experiments on synthetic and real-world data sets corroborate our claims and verify the robustness of the proposed framework.