Ultrahigh-dimensional Robust and Efficient Sparse Regression using Non-Concave Penalized Density Power Divergence

TL;DR

This paper introduces a robust, efficient sparse regression method suitable for ultrahigh-dimensional data, leveraging non-concave penalized density power divergence to improve robustness and theoretical guarantees over existing methods.

Contribution

It develops a novel sparse regression approach using non-concave penalties and density power divergence, with proven oracle properties and robustness in ultrahigh-dimensional settings.

Findings

Demonstrates robustness against data contamination.

Shows superior performance in simulations compared to existing methods.

Provides theoretical guarantees for estimator consistency and oracle properties.

Abstract

We propose a sparse regression method based on the non-concave penalized density power divergence loss function which is robust against infinitesimal contamination in very high dimensionality. Present methods of sparse and robust regression are based on $\ell_1$-penalization, and their theoretical properties are not well-investigated. In contrast, we use a general class of folded concave penalties that ensure sparse recovery and consistent estimation of regression coefficients. We propose an alternating algorithm based on the Concave-Convex procedure to obtain our estimate, and demonstrate its robustness properties using influence function analysis. Under some conditions on the fixed design matrix and penalty function, we prove that this estimator possesses large-sample oracle properties in an ultrahigh-dimensional regime. The performance and effectiveness of our proposed method for parameter estimation and prediction compared to state-of-the-art are demonstrated through simulation studies.