提出新方法应对输入分布不同时的函数型数据学习难题。
Towards regularized learning from functional data with covariate shift
- 用向量值再生核希尔伯特空间限制假设空间，构建可处理函数输出的算法。
- 理论证明了最优收敛速度，且在真实人脸图像数据上验证了鲁棒性。
- 通过集成多参数与多核估计器，自动选择调参，适合实际应用者。
本文研究在协变量偏移假设下，针对向量值回归的无监督域适应问题，提出一种通用正则化框架，基于向量值再生核希尔伯特空间（vRKHS）。当训练与测试数据的输入分布不一致时，会带来可靠学习的重大挑战。通过限制假设空间，我们设计了一种可处理函数输出的实用算子学习算法，并在一般源条件下的最优收敛率得到理论证明，为该场景下的正则化学习提供了理论基础。此外，提出一种基于聚合的策略，将不同正则化参数和不同核函数对应的估计器进行线性组合，解决了调参选择的关键难题，并给出了理论支持其有效性。最后，在真实人脸图像数据集上展示了该方法在缓解分布差异方面的鲁棒性和有效性。
原文摘要 · Abstract (English)
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.
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