This paper considers a high-dimensional linear regression problem where there are complex correlation structures among predictors. We propose a graph-constrained regularization procedure, named Sparse Laplacian Shrinkage with the Graphical Lasso Estimator (SLS-GLE). The procedure uses the estimated precision matrix to describe the specific information on the conditional dependence pattern among predictors, and encourages both sparsity on the regression model and the graphical model. We introduce the Laplacian quadratic penalty adopting the graph information, and give detailed discussions on the advantages of using the precision matrix to construct the Laplacian matrix. Theoretical properties and numerical comparisons are presented to show that the proposed method improves both model interpretability and accuracy of estimation. We also apply this method to a financial problem and prove that the proposed procedure is successful in assets selection.

本文考虑了一个高维线性回归问题，其中预测变量之间存在复杂的相关结构。我们提出了一种图约束正则化过程，称为带有图形套索估计器（SLS-GLE）的稀疏拉普拉斯收缩。该过程使用估计的精度矩阵来描述预测变量之间条件依赖模式的具体信息，并鼓励回归模型和图形模型的稀疏性。我们介绍了采用图信息的拉普拉斯二次惩罚，并详细讨论了使用精度矩阵构造拉普拉斯矩阵的优点。理论性质和数值比较表明所提出的方法提高了模型的可解释性和估计的准确性。