  This paper considers the partially functional linear model (PFLM) where all
predictive features consist of a functional covariate and a high dimensional
scalar vector. Over an infinite dimensional reproducing kernel Hilbert space,
the proposed estimation for PFLM is a least square approach with two mixed
regularizations of a function-norm and an $\ell_1$-norm. Our main task in this
paper is to establish the minimax rates for PFLM under high dimensional
setting, and the optimal minimax rates of estimation is established by using
various techniques in empirical process theory for analyzing kernel classes. In
addition, we propose an efficient numerical algorithm based on randomized
sketches of the kernel matrix. Several numerical experiments are implemented to
support our method and optimization strategy.
