The sparse Laplacian shrinkage estimator for high-dimensional regression

TL;DR

The paper introduces the sparse Laplacian shrinkage (SLS) method for high-dimensional regression, combining sparsity and smoothness penalties to improve variable selection and estimation in correlated predictor settings.

Contribution

It proposes a novel penalized estimator that incorporates predictor correlation patterns via Laplacian quadratic and demonstrates its oracle property in high-dimensional contexts.

Findings

SLS achieves selection consistency in high-dimensional settings.

Simulation results show SLS outperforms traditional methods.

Real data example illustrates practical effectiveness.

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. 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 >> 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 real data example is used to illustrate its application.