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This paper proposes scalable and fast algorithms for solving the Robust PCAproblem, namely recovering a low-rank matrix with an unknown fraction of itsentries being arbitrarily corrupted. This problem arises in many applications,such as image processing, web data ranking, and bioinformatic data analysis. Itwas recently shown that under surprisingly broad conditions, the Robust PCAproblem can be exactly solved via convex optimization that minimizes acombination of the nuclear norm and the $\ell^1$-norm . In this paper, we applythe method of augmented Lagrange multipliers (ALM) to solve this convexprogram. As the objective function is non-smooth, we show how to extend theclassical analysis of ALM to such new objective functions and prove theoptimality of the proposed algorithms and characterize their convergence rate.Empirically, the proposed new algorithms can be more than five times fasterthan the previous state-of-the-art algorithms for Robust PCA, such as theaccelerated proximal gradient (APG) algorithm. Moreover, the new algorithmsachieve higher precision, yet being less storage/memory demanding. We also showthat the ALM technique can be used to solve the (related but somewhat simpler)matrix completion problem and obtain rather promising results too. We furtherprove the necessary and sufficient condition for the inexact ALM to convergeglobally. Matlab code of all algorithms discussed are available athttp://perception.csl.illinois.edu/matrix-rank/home.html

The Augmented Lagrange Multiplier Method for Exact Recovery of Corrupted Low-Rank Matrices