The Sparse Laplacian Shrinkage Estimator for High-Dimensional Regression

• Activity Number: 483
• Type: Invited
• Date/Time: Wednesday, August 3, 2011 : 10:30 AM to 12:20 PM
• Sponsor: Section on Statistical Learning and Data Mining
• Author(s): Jian Huang*+ and Shuangge Ma and Hongzhe Li and Cun-Hui Zhang
• Companies: University of Iowa and Yale University and University of Pennsylvania and Rutgers University
• Keywords: Graphical structure ; Minimax concave penalty ; Penalized regression ; High-dimensional data ; Variable selection ; Oracle property

Abstract

We propose a new penalized method for variable selection and estimation that explicitly incorporates the correlation patterns among predictors. This method is based on a combination of the minimax concave penalty and Laplacian quadratic associated with a graph as the penalty function. We call it the sparse Laplacian shrinkage (SLS) method. The SLS uses the minimax concave penalty for encouraging sparsity and Laplacian quadratic penalty for promoting smoothness among coefficients associated with the correlated predictors. The SLS has a generalized grouping property with respect to the graph represented by the Laplacian quadratic. In a special case, it has a similar grouping property as the elastic net method. We show that the SLS possesses an oracle property in the sense that it is selection consistent and equal to the oracle Laplacian shrinkage estimator with high probability. This result holds in sparse, high-dimensional settings with $p \gg n$ under reasonable conditions. We derive a coordinate descent algorithm for computing the SLS estimates. Simulation studies are conducted to evaluate the performance of the SLS method and a data example is used to illustrate its application