A Generalized Adaptive Joint Learning Framework for High-Dimensional Time-Varying Models

In modern biomedical and econometric studies, longitudinal processes are often characterized by complex time-varying associations and abrupt regime shifts that are shared across correlated outcomes. Standard functional data analysis (FDA) methods, which prioritize smoothness, often fail to capture these dynamic structural features, particularly in high-dimensional settings. This article introduces Adaptive Joint Learning (AJL), a regularization framework designed to simultaneously perform functional variable selection and structural changepoint detection in multivariate time-varying coefficient models. We propose a convex optimization procedure that synergizes adaptive group-wise penalization with fused regularization, effectively borrowing strength across multiple outcomes to enhance estimation efficiency. We provide a rigorous theoretical analysis of the estimator in the ultra-high-dimensional regime (p >> n), establishing non-asymptotic error bounds and proving that AJL achieves the oracle property--performing as well as if the true active set and changepoint locations were known a priori. A key theoretical contribution is the explicit handling of approximation bias via undersmoothing conditions to ensure valid asymptotic inference. The proposed method is validated through comprehensive simulations and an application to Primary Biliary Cirrhosis (PBC) data. The analysis uncovers synchronized phase transitions in disease progression and identifies a parsimonious set of time-varying prognostic markers.