Open Access
November 2012 A General Theory of Concave Regularization for High-Dimensional Sparse Estimation Problems
Cun-Hui Zhang, Tong Zhang
Statist. Sci. 27(4): 576-593 (November 2012). DOI: 10.1214/12-STS399


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.


Download Citation

Cun-Hui Zhang. Tong Zhang. "A General Theory of Concave Regularization for High-Dimensional Sparse Estimation Problems." Statist. Sci. 27 (4) 576 - 593, November 2012.


Published: November 2012
First available in Project Euclid: 21 December 2012

zbMATH: 1331.62353
MathSciNet: MR3025135
Digital Object Identifier: 10.1214/12-STS399

Keywords: approximate solution , concave regularization , global solution , local solution , Oracle inequality , sparse recovery , Variable selection

Rights: Copyright © 2012 Institute of Mathematical Statistics

Vol.27 • No. 4 • November 2012
Back to Top