Modal additive models with data-driven structure identification

Additive models, due to their high flexibility, have received a great deal of attention in high dimensional regression analysis. Many efforts have been made on capturing interactions between predictive variables within additive models. However, typical approaches are designed based on conditional mean assumptions, which may fail to reveal the structure when data is contaminated by heavy-tailed noise. In this paper, we propose a penalized modal regression method, Modal Additive Models (MAM), based on a conditional mode assumption for simultaneous function estimation and structure identification. MAM approximates the non-parametric function through forward neural networks, and maximizes modal risk with constraints on the function space and group structure. The proposed approach can be implemented by the half-quadratic (HQ) optimization technique, and its asymptotic estimation and selection consistency are established. It turns out that MAM can achieve satisfactory learning rate and identify the target group structure with high probability. The effectiveness of MAM is also supported by some simulated examples.