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Abstract: We propose a first-order augmented Lagrangian algorithm FALC to solve thecomposite norm minimization problem min |sigmaFX-G| alpha + |CX- d| betasubject to AX-b in Q; where sigmaX denotes the vector of singular values ofX, the matrix norm |sigmaX| alpha denotes either the Frobenius, the nuclear,or the L2-operator norm of X, the vector norm |.| beta denotes either theL1-norm, L2-norm or the L infty-norm; Q is a closed convex set and A., C.,F. are linear operators from matrices to vector spaces of appropriatedimensions. Basis pursuit, matrix completion, robust principal componentpursuit PCP, and stable PCP problems are all special cases of the compositenorm minimization problem. Thus, FALC is able to solve all these problems in aunified manner. We show that any limit point of FALC iterate sequence is anoptimal solution of the composite norm minimization problem. We also show thatfor all epsilon > 0, the FALC iterates are epsilon-feasible and epsilon-optimalafter Olog1-epsilon iterations, which require O1-epsilon constrainedshrinkage operations and Euclidean projection onto the set Q. Surprisingly, onthe problem sets we tested, FALC required only Olog1-epsilon constrainedshrinkage, instead of the O1-epsilon worst case bound, to compute anepsilon-feasible and epsilon-optimal solution. To best of our knowledge, FALCis the first algorithm with a known complexity bound that solves the stable PCPproblem.



Autor: Necdet Serhat Aybat, Garud Iyengar

Fuente: https://arxiv.org/







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