Robust reduced-rank regression

• 成果类型: Article
• 署名作者: She, Y.; Chen, K.
• 署名单位: State University System of Florida; Florida State University; University of Connecticut
• 刊物名称: BIOMETRIKA
• ISSN/ISSBN: 0006-3444
• DOI: 10.1093/biomet/asx032
• 发表日期: 2017
• 页码: 633647
• 关键词: nuclear-norm penalization; generalized linear-models; variable selection; convex relaxation; Optimal Rates; estimators; algorithm; matrices

摘要

In high-dimensional multivariate regression problems, enforcing low rank in the coefficient matrix offers effective dimension reduction, which greatly facilitates parameter estimation and model interpretation. However, commonly used reduced-rank methods are sensitive to data corruption, as the low-rank dependence structure between response variables and predictors is easily distorted by outliers. We propose a robust reduced-rank regression approach for joint modelling and outlier detection. The problem is formulated as a regularized multivariate regression with a sparse mean-shift parameterization, which generalizes and unifies some popular robust multivariate methods. An efficient thresholding-based iterative procedure is developed for optimization. We show that the algorithm is guaranteed to converge and that the coordinatewise minimum point produced is statistically accurate under regularity conditions. Our theoretical investigations focus on non-asymptotic robust analysis, demonstrating that joint rank reduction and outlier detection leads to improved prediction accuracy. In particular, we show that redescending.-functions can essentially attain the minimax optimal error rate, and in some less challenging problems convex regularization guarantees the same low error rate. The performance of the proposed method is examined through simulation studies and real-data examples.