Sufficient Dimension Reduction via Inverse Conditional Mean or Variance Independence

2026-05-08

AI Summary

This study addresses the challenge of achieving effective sufficient dimension reduction in high-dimensional settings while preserving predictive information about the response variable. The authors propose a unified framework based on inverse conditional mean independence (ICMI) and inverse conditional variance independence (ICVI), systematically introducing these concepts to characterize the central subspace for the first time. They establish a theoretical link between inverse conditional moment independence and sufficient dimension reduction, and develop four classes of estimators that integrate projection and kernel methods to ensure both robustness and computational efficiency. Under high-dimensional asymptotics, the convergence rates of the proposed estimators are rigorously established. Simulation studies and real-data analyses demonstrate that the method exhibits strong robustness against outliers in the response and achieves superior dimension reduction performance.

Abstract

This paper presents a unified framework for sufficient dimension reduction (SDR) that generalizes several existing SDR techniques and offers new insights into the connection between inverse conditional moment independence and dimension reduction. The framework is built on two forms of inverse independence between the response vector and predictors: inverse conditional mean independence (ICMI) and inverse conditional variance independence (ICVI). For each form, we develop two general classes of matrices capable of recovering the central subspace, based on projection and kernel techniques respectively. This yields four distinct estimators: projection- and kernel-based variants under both ICMI and ICVI frameworks. Under standard regularity conditions, we establish the theoretical properties of these estimators and derive their convergence rates in high-dimensional settings. The proposed methods exhibit robustness to outliers in the response variable while maintaining computational competitiveness. Simulation studies and real-data analyses demonstrate the practical effectiveness of the proposed methods.

Problem

• Sufficient Dimension Reduction
• Inverse Conditional Mean Independence
• Inverse Conditional Variance Independence
• Central Subspace
• High-dimensional Settings

Innovation

• Sufficient Dimension Reduction
• Inverse Conditional Mean Independence
• Inverse Conditional Variance Independence
• Central Subspace
• Kernel Methods