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The stable principal component pursuit (SPCP) problem is a non-smooth convexoptimization problem, the solution of which has been shown both in theory andin practice to enable one to recover the low rank and sparse components of amatrix whose elements have been corrupted by Gaussian noise. In this paper, weshow how several fast first-order methods can be applied to this problem veryefficiently. Specifically, we show that the subproblems that arise whenapplying optimal gradient methods of Nesterov, alternating linearizationmethods and alternating direction augmented Lagrangian methods to the SPCPproblem either have closed-form solutions or have solutions that can beobtained with very modest effort. All but one of the methods analyzed requireat least one of the non-smooth terms in the objective function to be smoothedand obtain an eps-optimal solution to the SPCP problem in O(1/eps) iterations.The method that works directly with the fully non-smooth objective function, isproved to be convergent under mild conditions on the sequence of parameters ituses. Our preliminary computational tests show that the latter method, althoughits complexity is not known, is fastest and substantially outperforms existingmethods for the SPCP problem. To best of our knowledge, an algorithm for theSPCP problem that has O(1/eps) iteration complexity and has a per iterationcomplexity equal to that of a singular value decomposition is given for thefirst time.

Fast First-Order Methods for Stable Principal Component Pursuit