- Statist. Sci.
- Volume 27, Number 4 (2012), 576-593.
A General Theory of Concave Regularization for High-Dimensional Sparse Estimation Problems
Concave regularization methods provide natural procedures for sparse recovery. However, they are difficult to analyze in the high-dimensional setting. Only recently a few sparse recovery results have been established for some specific local solutions obtained via specialized numerical procedures. Still, the fundamental relationship between these solutions such as whether they are identical or their relationship to the global minimizer of the underlying nonconvex formulation is unknown. The current paper fills this conceptual gap by presenting a general theoretical framework showing that, under appropriate conditions, the global solution of nonconvex regularization leads to desirable recovery performance; moreover, under suitable conditions, the global solution corresponds to the unique sparse local solution, which can be obtained via different numerical procedures. Under this unified framework, we present an overview of existing results and discuss their connections. The unified view of this work leads to a more satisfactory treatment of concave high-dimensional sparse estimation procedures, and serves as a guideline for developing further numerical procedures for concave regularization.
Statist. Sci., Volume 27, Number 4 (2012), 576-593.
First available in Project Euclid: 21 December 2012
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Zhang, Cun-Hui; Zhang, Tong. A General Theory of Concave Regularization for High-Dimensional Sparse Estimation Problems. Statist. Sci. 27 (2012), no. 4, 576--593. doi:10.1214/12-STS399. https://projecteuclid.org/euclid.ss/1356098557
- Supplementary material: Supplementary material for “A general theory of concave regularization for high-dimensional sparse estimation problems”. Due to space considerations, the proofs in this paper are all given in the supplementary document .