# Towards regularized learning from functional data with covariate shift

arXiv: https://arxiv.org/abs/2601.21019 · HTML: https://mlsanity.com/paper/2601.21019/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 31 references that are in the map.
- First posted on arXiv 28 Jan 2026.

## Builds on (most cited first, 12 of 12)

- [Optimal Rates For Regularization Of Statistical Inverse Learning Problems](https://mlsanity.com/paper/1604.04054/) (2016)
- [Optimally tackling covariate shift in RKHS-based nonparametric regression](https://mlsanity.com/paper/2205.02986/) (2022)
- [Towards Optimal Sobolev Norm Rates for the Vector-Valued Regularized Least-Squares Algorithm](https://mlsanity.com/paper/2312.07186/) (2023)
- [Addressing Parameter Choice Issues in Unsupervised Domain Adaptation by Aggregation](https://mlsanity.com/paper/2305.01281/) (2023)
- [When is Importance Weighting Correction Needed for Covariate Shift Adaptation?](https://mlsanity.com/paper/2303.04020/) (2023)
- [Optimal Rates for Vector-Valued Spectral Regularization Learning Algorithms](https://mlsanity.com/paper/2405.14778/) (2024)
- [Optimal rates for the regularized learning algorithms under general source condition](https://mlsanity.com/paper/1611.01900/) (2016)
- [On regularized Radon-Nikodym differentiation](https://mlsanity.com/paper/2308.07887/) (2023)
- [Computational Efficiency under Covariate Shift in Kernel Ridge Regression](https://mlsanity.com/paper/2505.14083/) (2025)
- [Spectral Algorithms under Covariate Shift](https://mlsanity.com/paper/2504.12625/) (2025)
- [Spectral Algorithms in Misspecified Regression: Convergence under Covariate Shift](https://mlsanity.com/paper/2509.05106/) (2025)
- [Covariate-Adjusted Functional Data Analysis for Structural Health Monitoring](https://mlsanity.com/paper/2408.02106/) (2024)

## Built on by (most cited first, 1 of 1)

- [Scalable Operator Learning via Nystr"om Approximation With Denoising Applications](https://mlsanity.com/paper/2606.26652/) (2026)

## Abstract (arXiv)

This paper investigates a general regularization framework for unsupervised domain adaptation in vector-valued regression under the covariate shift assumption, utilizing vector-valued reproducing kernel Hilbert spaces (vRKHS). Covariate shift occurs when the input distributions of the training and test data differ, introducing significant challenges for reliable learning. By restricting the hypothesis space, we develop a practical operator learning algorithm capable of handling functional outputs. We establish optimal convergence rates for the proposed framework under a general source condition, providing a theoretical foundation for regularized learning in this setting. We also propose an aggregation-based approach that forms a linear combination of estimators corresponding to different regularization parameters and different kernels. The proposed approach addresses the challenge of selecting appropriate tuning parameters, which is crucial for constructing a good estimator, and we provide a theoretical justification for its effectiveness. Furthermore, we illustrate the proposed method on a real-world face image dataset, demonstrating robustness and effectiveness in mitigating distributional discrepancies under covariate shift.