Abstract

We study a high-dimensional nonparametric additive model where the response variable and the covariates take values in Hilbert manifolds. The spaces for the response variable and for the covariates are allowed to be distinct. In our framework, the number of the covariates may be larger than the sample size while significant covariates are sparse among them. We develop kernel smoothing techniques with various penalization schemes to estimate the additive component maps. We establish both weak and strong oracle properties and derive various types of non-asymptotic error bounds for the proposed estimators. Furthermore, we propose a computational algorithm that integrates the smooth backfitting projection theory with the alternating direction method of multipliers. The practical merit of our method is demonstrated through numerical simulations and real data analysis.