# Functional linear regression from sparse to dense designs: a pooling-ridge method and minimax optimality

arXiv: https://arxiv.org/abs/2608.25468 · HTML: https://mlsanity.com/paper/2608.25468/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 39 references that are in the map.
- First posted on arXiv 26 Aug 2026.

## Builds on (most cited first, 6 of 6)

- [On the Saturation Effect of Kernel Ridge Regression](https://mlsanity.com/paper/2405.09362/) (2024)
- [Low-Rank Covariance Function Estimation for Multidimensional Functional Data](https://mlsanity.com/paper/2008.12919/) (2020)
- [Optimal Penalized Function-on-Function Regression under a Reproducing Kernel Hilbert Space Framework](https://mlsanity.com/paper/1902.03674/) (2019)
- [Optimal Estimation for the Functional Cox Model](https://mlsanity.com/paper/1601.07496/) (2016)
- [Nonlinear function-on-function regression by RKHS](https://mlsanity.com/paper/2207.08211/) (2022)
- [Optimal estimation in functional linear regression for sparse noise-contaminated data](https://mlsanity.com/paper/1711.04854/) (2017)

## Abstract (arXiv)

Functional data analysis is an important statistical field that treats data as random functions. In practice, the random functions are often not fully observed but instead measured at discrete times. While simpler problems, such as mean and covariance estimation, have been widely studied for discretely observed data, optimal estimation of linear regression for this data type has remained unsolved for over two decades. To tackle this fundamental challenge, we propose a novel approach, referred to as pooling ridge estimation, which combines the advantages of pooling strategy and RKHS-based method by incorporating the unbiased estimation of operators based on discretely observed measurements from all subjects. This unified estimation framework enables us to achieve minimax optimality in prediction risk in arbitrary sampling schemes ranging from sparse to dense designs, for both scalar-on-function and function-on-function regression models. Such methodological and theoretical advances are obtained for the first time and accurately reveal the influence of discrete sampling. For scalar-on-function regression, the phase transition occurs once, separating the convergence behavior into two distinct regimes. Remarkably, for function-on-function regression, up to three phase transitions may occur, determined by the sampling frequencies of the predictor/response functions. Finally, simulation experiments and two real data examples provide empirical support for the proposed methods.