Partition-Based Functional Ridge Regression for High-Dimensional Data

2026-03-11

AI Summary

This study addresses the challenges of multicollinearity, overfitting, and interpretability in linear models for high-dimensional functional data by proposing a block-wise differential ridge regression framework based on coefficient function decomposition. The method partitions the coefficient function vector into dominant and weak-effect components and applies distinct ridge penalties to each, enabling adaptive regularization without explicit variable selection. Theoretical analysis establishes consistency and asymptotic normality for three novel estimators. Simulations demonstrate that the Functional Ridge Shrinkage Method (FRSM) substantially reduces variance in small samples, while the Functional Ridge Fusion Method (FRFM) enhances predictive accuracy in large samples. Application to Canadian weather data confirms the approach’s effectiveness in mitigating variance inflation, improving prediction performance, and clearly identifying key functional effects.

Technology Category

• Machine Learning: Dimensionality Reduction/Feature Selection
• Reasoning under Uncertainty: Applications
• Intelligent Robots: State Estimation

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization

Abstract

This paper proposes a partition-based functional ridge regression framework to address multicollinearity, overfitting, and interpretability in high-dimensional functional linear models. The coefficient function vector \( \boldsymbolβ(s) \) is decomposed into two components, \( \boldsymbolβ_1(s) \) and \( \boldsymbolβ_2(s) \), representing dominant and weaker functional effects. This partition enables differential ridge penalization across functional blocks, so that important signals are preserved while less informative components are more strongly shrunk. The resulting approach improves numerical stability and enhances interpretability without relying on explicit variable selection. We develop three estimators: the Functional Ridge Estimator (FRE), the Functional Ridge Full Model (FRFM), and the Functional Ridge Sub-Model (FRSM). Under standard regularity conditions, we establish consistency and asymptotic normality for all estimators. Simulation results reveal a clear bias--variance trade-off where FRSM performs best in small samples through strong variance reduction, whereas FRFM achieves superior accuracy in moderate to large samples by retaining informative functional structure through adaptive penalization. An empirical application to Canadian weather data further demonstrates improved predictive performance, reduced variance inflation, and clearer identification of influential functional effects. Overall, partition-based ridge regularization provides a practical and theoretically grounded method for high-dimensional functional regression.

Problem

• multicollinearity
• overfitting
• interpretability
• high-dimensional functional data
• functional linear models

Innovation

• partition-based regularization
• functional ridge regression
• high-dimensional functional data
• differential penalization
• interpretability