Concave regularization methods provide natural procedures for sparse recovery.  However, they are difficult to analyze in the high dimensional setting. In this talk I will first explain the advantage of non-convex regularization over Lasso, and review some recent results showing improved recovery performance for local solutions of nonconvex formulations obtained via specialized numerical procedures. I will then present a unified framework describing the relationship of these local minima to the global minimizer of the underlying nonconvex  formulation. In particular, we show that under suitable conditions, the global solution of nonconvex regularization leads to desirable recovery performance and it corresponds to the unique sparse local solution, which can  be obtained via different numerical procedures. This unified view leads to a more satisfactory treatment of concave high dimensional sparse estimation procedures, and can serve as the guideline for developing additional numerical procedures for concave regularization.

Joint work with Cunhui Zhang