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

Baolin Chen, Mengfei Ran

TL;DR

AJL tackles high-dimensional, multi-outcome longitudinal data with time-varying effects and shared regime shifts by unifying adaptive functional variable selection with changepoint detection. It casts time-varying intercepts and slopes into a B-spline based, ultra-high-dimensional framework and solves a convex objective that combines adaptive group lasso and adaptive fused penalties. Theoretical results establish non-asymptotic error bounds and an oracle property, with undersmoothing enabling valid pointwise inference. Empirical validation via extensive simulations and a PBC study demonstrates superior prediction, accurate changepoint localization, and interpretable time-varying prognostic markers, underscoring AJL’s potential for dynamic biomarker discovery in complex longitudinal data.

Abstract

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.

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

TL;DR

AJL tackles high-dimensional, multi-outcome longitudinal data with time-varying effects and shared regime shifts by unifying adaptive functional variable selection with changepoint detection. It casts time-varying intercepts and slopes into a B-spline based, ultra-high-dimensional framework and solves a convex objective that combines adaptive group lasso and adaptive fused penalties. Theoretical results establish non-asymptotic error bounds and an oracle property, with undersmoothing enabling valid pointwise inference. Empirical validation via extensive simulations and a PBC study demonstrates superior prediction, accurate changepoint localization, and interpretable time-varying prognostic markers, underscoring AJL’s potential for dynamic biomarker discovery in complex longitudinal data.

Abstract

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.
Paper Structure (39 sections, 8 theorems, 94 equations, 3 figures, 10 tables, 2 algorithms)

This paper contains 39 sections, 8 theorems, 94 equations, 3 figures, 10 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Methodology
  3. The Time-Varying Coefficient Model (TVCM)
  4. Basis Expansion for Functional Data
  5. Advantages of the Functional Basis Parameterization
  6. A Penalized Objective for Functional Data
  7. Theoretical Motivation for Adaptivity
  8. The Generalized Adaptive Joint Learning (AJL) Objective
  9. Optimization Algorithm
  10. Convergence and Complexity Analysis
  11. Theoretical Analysis
  12. Assumptions
  13. Non-Asymptotic Analysis
  14. Asymptotic Oracle Property
  15. Numerical Study
  16. ...and 24 more sections

Key Result

Proposition 1

Let $Q(\bm{A}, \{\bm{B}_j\})$ be the (adaptive) objective function in eq:adaptive_objective_final_func. The objective function value $Q(\widehat{\bm{A}}^{(s)}, \{\widehat{\bm{B}}_j^{(s)}\})$ is monotonically decreasing in $s$ and converges to a stationary point (a global minimum, as $Q$ is convex).

Figures (3)

  • Figure 1: Boxplots of $\mathrm{ISE}(\beta)$ across scenarios.
  • Figure 2: Boxplots of Prediction Error (PE) on the test set across eight simulation scenarios.
  • Figure 3: Analysis results on PBC data. Left: Estimated baseline trajectories (intercepts) for the three liver markers, showing a common trend change around $t=0.25$. Right: Estimated dynamic effects of selected covariates on Log-Bilirubin. Note that Treatment (green) is correctly identified as having zero effect.

Theorems & Definitions (19)

  • Remark 1: Implicit Robustness to Outliers
  • Remark 2: Hierarchical Regularization Strategy and Computational Strategy
  • Proposition 1: Convergence
  • proof
  • Remark 3: Adaptation and Relaxation
  • Lemma 1: Basic Inequality
  • Theorem 1: Estimation Error Rate of Generalized AJL
  • Theorem 2: Consistent Support Recovery & Structural Localization
  • Proposition 2: Validity of Hierarchical Screening
  • Remark 4
  • ...and 9 more