We study the transfer learning (TL) for the functional linear regression (FLR) under the Reproducing Kernel Hilbert Space (RKHS) framework, observing the TL techniques in existing high-dimensional linear regression is not compatible with the truncation-based FLR methods as functional data are intrinsically infinite-dimensional and generated by smooth underlying processes. We measure the similarity across tasks using RKHS distance, allowing the type of information being transferred tied to the properties of the imposed RKHS. Building on the hypothesis offset transfer learning paradigm, two algorithms are proposed: one conducts the transfer when positive sources are known, while the other leverages aggregation techniques to achieve robust transfer without prior information about the sources. We establish lower bounds for this learning problem and show the proposed algorithms enjoy a matching asymptotic upper bound. These analyses provide statistical insights into factors that contribute to the dynamics of the transfer. We also extend the results to functional generalized linear models. The effectiveness of the proposed algorithms is demonstrated on extensive synthetic data as well as a financial data application.

翻译：我们在再生核希尔伯特空间框架下研究泛函线性回归的迁移学习，指出现有高维线性回归中的迁移学习技术与基于截断的泛函线性回归方法不兼容，原因在于函数型数据本质上是无限维的且由平滑的底层过程生成。我们采用RKHS距离度量任务间的相似性，使得迁移信息的类型与所施加RKHS的性质相关联。基于假设偏移迁移学习范式，我们提出两种算法：一种在已知正源任务时进行迁移，另一种利用聚合技术，在无需源任务先验信息的情况下实现稳健迁移。我们建立了该学习问题的下界，并证明所提算法具有与之匹配的渐近上界。这些分析为理解迁移动力学的贡献因素提供了统计洞见。我们还将结果拓展至泛函广义线性模型。通过大量合成数据及金融数据应用验证了所提算法的有效性。