Sparse Regression Incorporating Graphical Structure Among Predictors

With the abundance of high-dimensional data in various disciplines, sparse regularized techniques are very popular these days. In this article, we make use of the structure information among predictors to improve sparse regression models. Typically, such structure information can be modeled by the connectivity of an undirected graph using all predictors as nodes of the graph. Most existing methods use this undirected graph edge-by-edge to encourage the regression coefficients of corresponding connected predictors to be similar. However, such methods do not directly use the neighborhood information of the graph. Furthermore, if there are more edges in the predictor graph, the corresponding regularization term will be more complicated. In this article, we incorporate the graph information node-by-node, instead of edge-by-edge as used in most existing methods. Our proposed method is very general' and it includes adaptive Lasso, group Lasso, and ridge regression as special cases. Both theoretical and numerical studies demonstrate the effectiveness of the proposed method for simultaneous estimation, prediction, and model selection. Supplementary materials for this article are available online.