Optimal prediction for kernel-based semi-functional linear regression

TL;DR

This paper establishes minimax optimal prediction rates for semi-functional linear models with both functional and nonparametric components, using a double-penalized least squares approach within RKHS.

Contribution

It introduces a novel double-penalized least squares method with an efficient, non-iterative algorithm for estimating both components in semi-functional linear models.

Findings

Achieves minimax optimal convergence rates for prediction.

Demonstrates the effectiveness of the proposed method through numerical studies.

Validates theoretical results with empirical evidence.

Abstract

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.