Sparse Laplacian Shrinkage with the Graphical Lasso Estimator for Regression Problems

TL;DR

This paper introduces a novel high-dimensional regression method that integrates graphical model information via the Graphical Lasso to enhance sparsity, interpretability, and estimation accuracy.

Contribution

The paper proposes SLS-GLE, a new regularization technique combining Laplacian penalties with the Graphical Lasso, improving model interpretability and estimation in complex predictor correlation structures.

Findings

Improves estimation accuracy over existing methods

Enhances interpretability of regression models

Successfully applied to financial asset selection

Abstract

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.