Sparse Reduced Rank Huber Regression in High Dimensions

• 成果类型: Article
• 署名作者: Tan, Kean Ming; Sun, Qiang; Witten, Daniela
• 署名单位: University of Michigan System; University of Michigan; University of Toronto; University of Washington; University of Washington Seattle
• 刊物名称: JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION
• ISSN/ISSBN: 0162-1459
• DOI: 10.1080/01621459.2022.2050243
• 发表日期: 2023
• 页码: 2383-2393
• 关键词: robust regression; gene network; M-ESTIMATORS; asymptotics; selection

摘要

We propose a sparse reduced rank Huber regression for analyzing large and complex high-dimensional data with heavy-tailed random noise. The proposed method is based on a convex relaxation of a rank-and sparsity-constrained nonconvex optimization problem, which is then solved using a block coordinate descent and an alternating direction method of multipliers algorithm. We establish nonasymptotic estimation error bounds under both Frobenius and nuclear norms in the high-dimensional setting. This is a major contribution over existing results in reduced rank regression, which mainly focus on rank selection and prediction consistency. Our theoretical results quantify the tradeoff between heavy-tailedness of the random noise and statistical bias. For random noise with bounded (1 + delta)th moment with delta is an element of (0, 1), the rate of convergence is a function of delta, and is slower than the sub-Gaussian-type deviation bounds; for random noise with bounded second moment, we obtain a rate of convergence as if sub-Gaussian noise were assumed. We illustrate the performance of the proposed method via extensive numerical studies and a data application. Supplementary materials for this article are available online.