Rank-Based Sparse Regression in Principal Components Space under Measurement Error

TL;DR

This paper introduces a robust rank-based sparse regression method in principal components space that effectively handles measurement errors and heavy-tailed noise, improving stability and prediction accuracy.

Contribution

It proposes a novel rank-based approach with adaptive reweighting to enhance robustness against heavy-tailed errors and predictor contamination in high-dimensional regression.

Findings

The rank-based method is competitive under Gaussian noise.

It is substantially more stable under heavy-tailed errors.

The procedure effectively handles predictor contamination.

Abstract

We study high-dimensional regression in principal components space when the predictors are observed with additive measurement error and the response errors may be heavy-tailed. The starting point is the $\ell_1$-penalized principal-components estimator of Song and Zou (2026), which enjoys a blessing-of-dimensionality phenomenon under predictor contamination but senstive for heavy-tailed data or outliers. We replace the squared loss by a Wilcoxon-type rank loss and then apply a one-step adaptive reweighting scheme to reduce the shrinkage bias of the initial $\ell_1$ fit. The resulting procedure combines robustness to heavy-tailed response errors with the contamination geometry induced by the empirical principal-components basis. Our main theorem gives a prediction bound for the fixed-$\lambda$ second-stage fitted mean. Simulations show that the rank-based procedure is competitive under Gaussian noise and substantially more stable under heavy-tailed errors, especially when predictor contamination is present.