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-$λ$ 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.

翻译：我们研究当预测变量观测值包含加性测量误差且响应误差可能具有重尾分布时，主成分空间中的高维回归问题。研究的出发点是Song与Zou（2026）提出的$\ell_1$惩罚主成分估计量，该估计量在预测变量受到污染时展现出"维度的祝福"现象，但对重尾数据或异常值较为敏感。我们将平方损失替换为Wilcoxon型秩损失，然后应用一步自适应重加权方案以减少初始$\ell_1$拟合的收缩偏差。由此得到的程序兼具对重尾响应误差的稳健性，以及由经验主成分基所诱导的污染几何结构。我们的主要定理给出了固定$\lambda$第二阶段拟合均值的预测界。模拟结果表明，在高斯噪声环境下，基于秩的方法具有竞争力；而在重尾误差下，尤其是存在预测变量污染时，该方法显著更为稳定。