ROP: MATRIX RECOVERY VIA RANK-ONE PROJECTIONS

• 成果类型: Article
• 署名作者: Cai, T. Tony; Zhang, Anru
• 署名单位: University of Pennsylvania
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/14-AOS1267
• 发表日期: 2015
• 页码: 102-138
• 关键词: sparse signals; Optimal Rates; PCA

摘要

Estimation of low-rank matrices is of significant interest in a range of contemporary applications. In this paper, we introduce a rank-one projection model for low-rank matrix recovery and propose a constrained nuclear norm minimization method for stable recovery of low-rank matrices in the noisy case. The procedure is adaptive to the rank and robust against small perturbations. Both upper and lower bounds for the estimation accuracy under the Frobenius norm loss are obtained. The proposed estimator is shown to be rate-optimal under certain conditions. The estimator is easy to implement via convex programming and performs well numerically. The techniques and main results developed in the paper also have implications to other related statistical problems. An application to estimation of spiked covariance matrices from one-dimensional random projections is considered. The results demonstrate that it is still possible to accurately estimate the covariance matrix of a high-dimensional distribution based only on one-dimensional projections.