2011/08/25 by Cun-Hui Zhang, Cun‐Hui Zhang, Tong Zhang +2 · 1 citation
Engineering · Mathematics · #Advanced Optimization Algorithms Research #FOS: Computer and information sciences #Machine Learning (stat.ML) #Numerical methods in inverse problems #Sparse and Compressive Sensing Techniques #stat.ML
paper · pdf · doi:10.48550/arxiv.1108.4988
30 pages
openalex publication_date 2011/08/25 · arxiv created 2012/02/11 · arxiv updated 2012/02/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 guideline for developing further numerical procedures for concave regularization.