Structure Adaptive Lasso
Lasso is of fundamental importance in high dimensional statistics and has been routinely used to regress a response on a high dimensional set of predictors. In many scientific applications, there exists external information that encodes the predictive power and sparsity structure of the predictors. In this article, we develop a new method, called the Structure Adaptive Lasso (SA Lasso) to incorporate these potentially useful side information into a penalized regression. The basic idea is to translate the external information into different penalization strengths for the regression coefficients. We study the risk properties of the resulting estimator. In particular, we generalize the state evolution framework recently introduced for the analysis of the approximate message passing algorithm to the SA Lasso setting. We show that the finite sample risk of the SA Lasso estimator is consistent with the theoretical risk predicted by the state evolution equation. Our theory suggests that the SA Lasso with an informative group or covariate structure can significantly outperform the Lasso, Adaptive Lasso, and Sparse Group Lasso. This evidence is further confirmed in our numerical studies. We also demonstrate the usefulness and the superiority of our method in a real data application.