  We study a functional linear regression model that deals with functional
responses and allows for both functional covariates and high-dimensional vector
covariates. The proposed model is flexible and nests several functional
regression models in the literature as special cases. Based on the theory of
reproducing kernel Hilbert spaces (RKHS), we propose a penalized least squares
estimator that can accommodate functional variables observed on discrete sample
points. Besides a conventional smoothness penalty, a group Lasso-type penalty
is further imposed to induce sparsity in the high-dimensional vector
predictors. We derive finite sample theoretical guarantees and show that the
excess prediction risk of our estimator is minimax optimal. Furthermore, our
analysis reveals an interesting phase transition phenomenon that the optimal
excess risk is determined jointly by the smoothness and the sparsity of the
functional regression coefficients. A novel efficient optimization algorithm
based on iterative coordinate descent is devised to handle the smoothness and
group penalties simultaneously. Simulation studies and real data applications
illustrate the promising performance of the proposed approach compared to the
state-of-the-art methods in the literature.
