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Concave regularization methods provide natural procedures for sparserecovery. However, they are difficult to analyze in the high dimensionalsetting. Only recently a few sparse recovery results have been established forsome specific local solutions obtained via specialized numerical procedures.Still, the fundamental relationship between these solutions such as whetherthey are identical or their relationship to the global minimizer of theunderlying nonconvex formulation is unknown. The current paper fills thisconceptual gap by presenting a general theoretical framework showing that underappropriate conditions, the global solution of nonconvex regularization leadsto desirable recovery performance; moreover, under suitable conditions, theglobal solution corresponds to the unique sparse local solution, which can beobtained via different numerical procedures. Under this unified framework, wepresent an overview of existing results and discuss their connections. Theunified view of this work leads to a more satisfactory treatment of concavehigh dimensional sparse estimation procedures, and serves as guideline fordeveloping further numerical procedures for concave regularization.

A General Theory of Concave Regularization for High Dimensional Sparse Estimation Problems