The Annals of Statistics

ROP: Matrix recovery via rank-one projections

T. Tony Cai and Anru Zhang

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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.

Article information

Ann. Statist., Volume 43, Number 1 (2015), 102-138.

First available in Project Euclid: 18 November 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62H12: Estimation
Secondary: 62H36 62C20: Minimax procedures

Constrained nuclear norm minimization low-rank matrix recovery optimal rate of convergence rank-one projection restricted uniform boundedness spiked covariance matrix


Cai, T. Tony; Zhang, Anru. ROP: Matrix recovery via rank-one projections. Ann. Statist. 43 (2015), no. 1, 102--138. doi:10.1214/14-AOS1267.

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Supplemental materials

  • Supplementary material: Supplement to “ROP: Matrix recovery via rank-one projections”. We prove the technical lemmas used in the proofs of the main results in this supplement. The proofs rely on results in [7, 13, 28, 36, 39, 41] and [31].