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February 2015 ROP: Matrix recovery via rank-one projections
T. Tony Cai, Anru Zhang
Ann. Statist. 43(1): 102-138 (February 2015). DOI: 10.1214/14-AOS1267

Abstract

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.

Citation

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T. Tony Cai. Anru Zhang. "ROP: Matrix recovery via rank-one projections." Ann. Statist. 43 (1) 102 - 138, February 2015. https://doi.org/10.1214/14-AOS1267

Information

Published: February 2015
First available in Project Euclid: 18 November 2014

zbMATH: 1308.62120
MathSciNet: MR3285602
Digital Object Identifier: 10.1214/14-AOS1267

Subjects:
Primary: 62H12
Secondary: 62C20, 62H36

Rights: Copyright © 2015 Institute of Mathematical Statistics

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Vol.43 • No. 1 • February 2015
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