Partially linear models (PLMs) are important generalizations of linear models and are very useful for analyzing high-dimensional data. Compared to linear models, the PLMs possess desirable flexibility of non-parametric regression models because they have both linear and non-linear components. Variable selection for PLMs plays an important role in practical applications and has been extensively studied with respect to the linear component. However, for the non-linear component, variable selection has been well developed only for PLMs with extra structural assumptions such as additive PLMs and generalized additive PLMs. There is currently an unmet need for variable selection methods applicable to general PLMs without structural assumptions on the non-linear component. In this paper, we propose a new variable selection method based on learning gradients for general PLMs without any assumption on the structure of the non-linear component. The proposed method utilizes the reproducing-kernel-Hilbert-space tool to learn the gradients and the group-lasso penalty to select variables. In addition, a block-coordinate descent algorithm is suggested and some theoretical properties are established including selection consistency and estimation consistency. The performance of the proposed method is further evaluated via simulation studies and illustrated using real data.
"Variable selection for partially linear models via learning gradients." Electron. J. Statist. 11 (2) 2907 - 2930, 2017. https://doi.org/10.1214/17-EJS1300