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

arXiv: https://arxiv.org/abs/1902.03674 · HTML: https://mlsanity.com/paper/1902.03674/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Feb 2019.

## Built on by (most cited first, 5 of 5)

- [Modern Non-Linear Function-on-Function Regression](https://mlsanity.com/paper/2107.14151/) (2021)
- [On Hypothesis Transfer Learning of Functional Linear Models](https://mlsanity.com/paper/2206.04277/) (2022)
- [FAME: Adaptive Functional Attention with Expert Routing for Function-on-Function Regression](https://mlsanity.com/paper/2510.00621/) (2025)
- [Functional linear regression from sparse to dense designs: a pooling-ridge method and minimax optimality](https://mlsanity.com/paper/2608.25468/) (2026)
- [Functional Attentive Interpretable Regression](https://mlsanity.com/paper/2609.05846/) (2026)

## Abstract (arXiv)

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.