Robust selection and estimation for sparse multivariate functional nonparametric additive models via regularized Huber regression

In this paper, we investigate sparse functional additive models with multivariate functional predictors and a scalar response variable. This model adopts a nonparametric additive framework to flexibly incorporate multivariate functional principal component analysis (FPCA) scores, effectively capturing complex nonlinear relationships while mitigating the curse of dimensionality. To enhance robustness against outliers and heavy-tailed errors, we propose a regularized Huber regression method incorporating the component selection and smoothing operator (COSSO) penalty. The proposed approach is formulated within a reproducing kernel Hilbert space (RKHS) framework, enabling simultaneous component selection and estimation in a robust manner. Furthermore, we extend the locally adaptive majorize-minimization (LAMM) principle to develop a general iterative optimization algorithm applicable to any loss function with continuous gradients. Under mild assumptions on the error distribution (without requiring sub-Gaussian tails) and standard regularity conditions, we establish theoretical guarantees for the proposed estimator. Extensive simulation studies and a real data application to fluorescence spectroscopy demonstrate the superior performance of our method compared to existing alternatives.