Kernel-based estimation for partially functional linear model: Minimax rates and randomized sketches

TL;DR

This paper develops a minimax optimal kernel-based estimation method for the partially functional linear model in high dimensions, incorporating regularization and randomized sketching for efficiency, supported by theoretical analysis and numerical experiments.

Contribution

It introduces a new kernel-based estimation approach for PFLM with proven minimax rates and an efficient randomized sketching algorithm, advancing high-dimensional functional data analysis.

Findings

Established minimax rates for PFLM estimation.

Proposed an efficient randomized sketching algorithm.

Validated the method through numerical experiments.

Abstract

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.