A unified approach for minimizing composite norms

We propose a first-order augmented Lagrangian algorithm (FALC) to solve the composite norm minimizationproblemminX∈Rm×n μ1‖σ(F(X)−G)‖α +μ2‖C(X)− d‖β ,subject to A(X)− b ∈ Q,where σ(X) denotes the vector of singular values of X ∈ Rm×n, the matrix norm‖σ(X)‖α denotes either the Frobenius,the nuclear, or the `2-operator norm of X, the vector norm‖.‖β denotes either the `1-norm, `2-norm or the `∞-norm; Q isa closed convex set and A(.), C(.), F(.) are linear operators from Rm×n to vector spaces of appropriate dimensions. Basispursuit, matrix completion, robust principal component pursuit (PCP), and stable PCP problems are all special cases of thecomposite norm minimization problem. Thus, FALC is able to solve all these problems in a unified manner. We show thatany limit point of FALC iterate sequence is an optimal solution of the composite norm minimization problem. We also showthat for all > 0, the FALC iterates are -feasible and -optimal after O(log( −1)) iterations, which require O( −1) constrainedshrinkage operations and Euclidean projection onto the set Q. Surprisingly, on the problem sets we tested, FALC required onlyO(log( −1)) constrained shrinkage, instead of the O( −1) worst case bound, to compute an -feasible and -optimal solution.To best of our knowledge, FALC is the first algorithm with a known complexity bound that solves the stable PCP problem.