Optimal Rates for Functional Linear Regression with General Regularization

TL;DR

This paper derives optimal convergence rates for functional linear regression using spectral regularization within reproducing kernel Hilbert spaces, advancing theoretical understanding of estimation accuracy under smoothness assumptions.

Contribution

It introduces a general spectral regularization framework for functional linear regression and establishes sharp convergence rates, extending previous results in the field.

Findings

Optimal convergence rates for estimation and prediction errors.

Generalization of existing results under broader smoothness conditions.

Sharp theoretical bounds for spectral regularization methods.

Abstract

Functional linear regression is one of the fundamental and well-studied methods in functional data analysis. In this work, we investigate the functional linear regression model within the context of reproducing kernel Hilbert space by employing general spectral regularization to approximate the slope function with certain smoothness assumptions. We establish optimal convergence rates for estimation and prediction errors associated with the proposed method under a H\"{o}lder type source condition, which generalizes and sharpens all the known results in the literature.