Towards regularized learning from functional data with covariate shift
This addresses the challenge of reliable learning when training and test data have different input distributions, which is an incremental advance in domain adaptation methods.
The paper tackles the problem of unsupervised domain adaptation for vector-valued regression under covariate shift by developing a regularization framework in vector-valued reproducing kernel Hilbert spaces. It establishes optimal convergence rates, proposes an aggregation method for tuning parameter selection, and demonstrates effectiveness on a face image dataset with robustness to distributional discrepancies.
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.