GEOMETRIC INFERENCE FOR GENERAL HIGH-DIMENSIONAL LINEAR INVERSE PROBLEMS

• 成果类型: Article
• 署名作者: Cai, T. Tony; Liang, Tengyuan; Rakhlin, Alexander
• 署名单位: University of Pennsylvania
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/15-AOS1426
• 发表日期: 2016
• 页码: 1536-1563
• 关键词: statistical estimation; confidence-intervals; DANTZIG SELECTOR; matrix; Penalization; Lasso; rates

摘要

This paper presents a unified geometric framework for the statistical analysis of a general ill-posed linear inverse model which includes as special cases noisy compressed sensing, sign vector recovery, trace regression, orthogonal matrix estimation and noisy matrix completion. We propose computationally feasible convex programs for statistical inference including estimation, confidence intervals and hypothesis testing. A theoretical framework is developed to characterize the local estimation rate of convergence and to provide statistical inference guarantees. Our results are built based on the local conic geometry and duality. The difficulty of statistical inference is captured by the geometric characterization of the local tangent cone through the Gaussian width and Sudakov estimate.