The sparse laplacian shrinkage estimator for high-dimensional regression

Jian Huang, Shuangge Ma, Hongzhe Li, Cun Hui Zhang

Research output: Contribution to journalArticle

55 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)2021-2046
Number of pages26
JournalAnnals of Statistics
Issue number4
StatePublished - Aug 1 2011

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


  • Graphical structure
  • High-dimensional data
  • Minimax concave penalty
  • Oracle property.
  • Penalized regression
  • Variable selection

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