The additive model is one of the most popularly used models for high dimensional nonparametric regression analysis. However, its main drawback is that it neglects possible interactions between predictor variables. In this paper, we reexamine the group additive model proposed in the literature, and rigorously define the intrinsic group additive structure for the relationship between the response variable $Y$ and the predictor vector $\vect{X}$, and further develop an effective structure-penalized kernel method for simultaneous identification of the intrinsic group additive structure and nonparametric function estimation. The method utilizes a novel complexity measure we derive for group additive structures. We show that the proposed method is consistent in identifying the intrinsic group additive structure. Simulation study and real data applications demonstrate the effectiveness of the proposed method as a general tool for high dimensional nonparametric regression.

The paper considers kernel regression in the high dimensional setting. The starting point of the paper is the "additive model", which consider regressors on the form f sum_p f_p(X_p) i.e. correlations between different dimensions are ignored. Obviously it is problematic to ignore such correlations so the authors instead consider the "group additive model", where different dimensions are grouped, such that correlations between some dimensions can be modeled. The authors provide a formal treatment of this setting, and provide some (somewhat) ad hoc algorithms for finding optimal groupings. Results are presented for small synthetic and real-world cases.

The additive model is one of the most popularly used models for high dimensional nonparametric regression analysis. However, its main drawback is that it neglects possible interactions between predictor variables. In this paper, we reexamine the group additive model proposed in the literature, and rigorously define the intrinsic group additive structure for the relationship between the response variable $Y$ and the predictor vector $\vect{X}$, and further develop an effective structure-penalized kernel method for simultaneous identification of the intrinsic group additive structure and nonparametric function estimation. The method utilizes a novel complexity measure we derive for group additive structures. We show that the proposed method is consistent in identifying the intrinsic group additive structure.