Kernel-based estimation for partially functional linear model: Minimax rates and randomized sketches
This work addresses statistical estimation in high-dimensional functional data models, which is an incremental contribution to the field of functional data analysis.
The paper tackles the partially functional linear model (PFLM) with functional and high-dimensional scalar covariates by proposing a least squares estimator with mixed regularization, establishing minimax rates for estimation under high-dimensional settings, and developing an efficient algorithm using randomized sketches of the kernel matrix. The results include theoretical minimax rates and numerical experiments to validate the method.
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.