# Optimal Penalized Function-on-Function Regression under a Reproducing Kernel Hilbert Space Framework

## TL;DR

This paper introduces a penalized function-on-function regression model within a Reproducing Kernel Hilbert Space framework, providing a data-adaptive finite-dimensional solution with proven minimax convergence and demonstrated advantages through simulations and real data applications.

## Contribution

It develops a novel penalized regression estimator for function-on-function data with a Representer Theorem, enabling efficient optimization and theoretical guarantees.

## Key findings

- Estimator achieves minimax convergence rate in mean prediction.
- Simulation studies show numerical advantages over existing methods.
- Application to real data demonstrates practical utility.

## Abstract

Many scientific studies collect data where the response and predictor variables are both functions of time, location, or some other covariate. Understanding the relationship between these functional variables is a common goal in these studies. Motivated from two real-life examples, we present in this paper a function-on-function regression model that can be used to analyze such kind of functional data. Our estimator of the 2D coefficient function is the optimizer of a form of penalized least squares where the penalty enforces a certain level of smoothness on the estimator. Our first result is the Representer Theorem which states that the exact optimizer of the penalized least squares actually resides in a data-adaptive finite dimensional subspace although the optimization problem is defined on a function space of infinite dimensions. This theorem then allows us an easy incorporation of the Gaussian quadrature into the optimization of the penalized least squares, which can be carried out through standard numerical procedures. We also show that our estimator achieves the minimax convergence rate in mean prediction under the framework of function-on-function regression. Extensive simulation studies demonstrate the numerical advantages of our method over the existing ones, where a sparse functional data extension is also introduced. The proposed method is then applied to our motivating examples of the benchmark Canadian weather data and a histone regulation study.