ROP: Matrix Recovery via Rank-One Projections

• Activity Number: 558
• Type: Contributed
• Date/Time: Wednesday, August 6, 2014 : 2:00 PM to 3:50 PM
• Sponsor: IMS
• Author(s): Anru Zhang*+ and Tony Cai
• Companies: University of Pennsylvania and University of Pennsylvania
• Keywords: constrained nuclear norm minimization ; low-rank matrix recovery ; optimal rate of convergence ; rank-one projection ; restricted uniform boundedness ; spike covariance matrix

Abstract

Estimation of low-rank matrices is of signicant interest in a range of contemporary applications. In this talk, 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 low-rank 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 main results also have implications to other related statistical problems. An application to estimation of spike covariance matrices from one-dimensional random projections is considered. The results demonstrate that it is possible to accurately estimate the covariance matrix of a high-dimensional distribution based only on one-dimensional projections.

This is the joint work with T. Tony Cai.