Coefficient Shape Transfer Learning for Functional Linear Regression

Abstract

The shapes of functions provide highly interpretable summaries of their trajectories. This article develops a novel transfer learning methodology to tackle the challenge of data scarcity in functional linear models. The methodology incorporates samples from the target model (target domain) alongside those from auxiliary models (source domains), transferring knowledge of coefficient shape from the source domains to the target domain. This shape-based transfer learning framework enhances robustness and generalizability: by being invariant to covariate scaling and signal strength, it ensures reliable knowledge transfer even when data from different sources differ in magnitude, and by formalizing the notion of coefficient shape homogeneity, it extends beyond traditional coefficient-equality assumptions to incorporate information from a broader range of source domains. We rigorously analyze the convergence rates of the proposed estimator and examine the minimax optimality. Our findings show that the degree of improvement depends not only on the similarity of coefficient shapes between the target and source domains, but also on coefficient magnitudes and the spectral decay rates of the functional covariates covariance operators. To address situations where only a subset of auxiliary models is informative for the target model, we further develop a data-driven procedure for identifying such informative sources. The effectiveness of the proposed methodology is demonstrated through comprehensive simulation studies and an application to occupation time analysis using physical activity data from the U.S. National Health and Nutrition Examination Survey.