A reproducing kernel Hilbert space approach to functional linear regression

TL;DR

This paper introduces a smoothness regularization method for functional linear regression using reproducing kernel Hilbert spaces, achieving optimal convergence rates and demonstrating practical implementability.

Contribution

It develops a unified framework for prediction and estimation in functional linear regression using kernel diagonalization, improving upon existing methods.

Findings

Achieves minimax optimal convergence rates

Method is easily implementable in practice

Numerical results validate theoretical advantages

Abstract

We study in this paper a smoothness regularization method for functional linear regression and provide a unified treatment for both the prediction and estimation problems. By developing a tool on simultaneous diagonalization of two positive definite kernels, we obtain shaper results on the minimax rates of convergence and show that smoothness regularized estimators achieve the optimal rates of convergence for both prediction and estimation under conditions weaker than those for the functional principal components based methods developed in the literature. Despite the generality of the method of regularization, we show that the procedure is easily implementable. Numerical results are obtained to illustrate the merits of the method and to demonstrate the theoretical developments.