Robust High-Dimensional Regression with Coefficient Thresholding and its Application to Imaging Data Analysis

TL;DR

This paper introduces a robust high-dimensional regression method using coefficient thresholding and Huber loss, effectively handling complex dependence and outliers, with theoretical guarantees and real-world imaging data application.

Contribution

It develops a novel nonconvex estimation procedure for high-dimensional data that is robust and accounts for predictor dependence, with rigorous theoretical analysis and practical validation.

Findings

Method achieves statistical consistency and convergence.

Demonstrates robustness against outliers.

Successfully applied to imaging data analysis.

Abstract

It is of importance to develop statistical techniques to analyze high-dimensional data in the presence of both complex dependence and possible outliers in real-world applications such as imaging data analyses. We propose a new robust high-dimensional regression with coefficient thresholding, in which an efficient nonconvex estimation procedure is proposed through a thresholding function and the robust Huber loss. The proposed regularization method accounts for complex dependence structures in predictors and is robust against outliers in outcomes. Theoretically, we analyze rigorously the landscape of the population and empirical risk functions for the proposed method. The fine landscape enables us to establish both {statistical consistency and computational convergence} under the high-dimensional setting. The finite-sample properties of the proposed method are examined by extensive simulation studies. An illustration of real-world application concerns a scalar-on-image regression analysis for an association of psychiatric disorder measured by the general factor of psychopathology with features extracted from the task functional magnetic resonance imaging data in the Adolescent Brain Cognitive Development study.