  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.
