  The stable principal component pursuit (SPCP) problem is a non-smooth convex
optimization problem, the solution of which has been shown both in theory and
in practice to enable one to recover the low rank and sparse components of a
matrix whose elements have been corrupted by Gaussian noise. In this paper, we
show how several fast first-order methods can be applied to this problem very
efficiently. Specifically, we show that the subproblems that arise when
applying optimal gradient methods of Nesterov, alternating linearization
methods and alternating direction augmented Lagrangian methods to the SPCP
problem either have closed-form solutions or have solutions that can be
obtained with very modest effort. All but one of the methods analyzed require
at least one of the non-smooth terms in the objective function to be smoothed
and 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, is
proved to be convergent under mild conditions on the sequence of parameters it
uses. Our preliminary computational tests show that the latter method, although
its complexity is not known, is fastest and substantially outperforms existing
methods for the SPCP problem. To best of our knowledge, an algorithm for the
SPCP problem that has O(1/eps) iteration complexity and has a per iteration
complexity equal to that of a singular value decomposition is given for the
first time.
