  In this paper, we establish minimax optimal rates of convergence for
prediction in a semi-functional linear model that consists of a functional
component and a less smooth nonparametric component. Our results reveal that
the smoother functional component can be learned with the minimax rate as if
the nonparametric component were known. More specifically, a double-penalized
least squares method is adopted to estimate both the functional and
nonparametric components within the framework of reproducing kernel Hilbert
spaces. By virtue of the representer theorem, an efficient algorithm that
requires no iterations is proposed to solve the corresponding optimization
problem, where the regularization parameters are selected by the generalized
cross validation criterion. Numerical studies are provided to demonstrate the
effectiveness of the method and to verify the theoretical analysis.
