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Necdet Serhat Aybat (nsa2106columbia.edu) Abstract: We propose a firstorder augmented Lagrangian algorithm (FALC) to solve the composite norm minimization problem min sigma(F(X)G)_alpha + C(X) d_beta subject to A(X)b in Q; where sigma(X) denotes the vector of singular values of X, the matrix norm sigma(X)_alpha denotes either the Frobenius, the nuclear, or the L2operator norm of X, the vector norm ._beta denotes either the L1norm, L2norm or the L inftynorm; Q is a closed convex set and A(.), C(.), F(.) are linear operators from matrices to vector spaces of appropriate dimensions. Basis pursuit, matrix completion, robust principal component pursuit (PCP), and stable PCP problems are all special cases of the composite norm minimization problem. Thus, FALC is able to solve all these problems in a unified manner. We show that any limit point of FALC iterate sequence is an optimal solution of the composite norm minimization problem. We also show that for all epsilon > 0, the FALC iterates are epsilonfeasible and epsilonoptimal after O(log(1/epsilon)) iterations, which require O(1/epsilon) constrained shrinkage operations and Euclidean projection onto the set Q. Surprisingly, on the problem sets we tested, FALC required only O(log(1/epsilon)) constrained shrinkage, instead of the O(1/epsilon) worst case bound, to compute an epsilonfeasible and epsilonoptimal solution. To best of our knowledge, FALC is the first algorithm with a known complexity bound that solves the stable PCP problem. Keywords: Augmented Lagrangian Method, First Order Method, Matrix Completion, Nuclear Norm, Robust PCA, Compressed Sensing, Basis Pursuit Category 1: Convex and Nonsmooth Optimization Category 2: Convex and Nonsmooth Optimization (Nonsmooth Optimization ) Citation: Download: [PDF] Entry Submitted: 05/25/2010 Modify/Update this entry  
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