Minimax Prediction for Functional Linear Regression with Functional Responses in Reproducing Kernel Hilbert Spaces

TL;DR

This paper establishes convergence rates for functional linear regression with functional responses in RKHS without requiring kernel alignment or eigenvalue decay assumptions, and demonstrates improved prediction performance over functional PCA through simulations and benchmarks.

Contribution

It provides the first convergence rate results under minimal assumptions and compares favorably to existing methods in practical datasets.

Findings

Proposed method outperforms functional PCA in prediction accuracy.

Convergence rates are established without kernel alignment or eigenvalue decay assumptions.

Simulation and benchmark results validate the effectiveness of the approach.

Abstract

In this article, we consider convergence rates in functional linear regression with functional responses, where the linear coefficient lies in a reproducing kernel Hilbert space (RKHS). Without assuming that the reproducing kernel and the covariate covariance kernel are aligned, or assuming polynomial rate of decay of the eigenvalues of the covariance kernel, convergence rates in prediction risk are established. The corresponding lower bound in rates is derived by reducing to the scalar response case. Simulation studies and two benchmark datasets are used to illustrate that the proposed approach can significantly outperform the functional PCA approach in prediction.