• 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 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 obtained in the paper 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.

     

  • Paper: pdf file.

  • Supplement: This supplement contains additional technical proofs.

     

  • Other related papers:

    Cai, T. & Zhang, A. (2014).
    Sparse representation of a polytope and recovery of sparse signals and low-rank matrices.
    IEEE Transactions on Information Theory 60, 122-132.

    Cai, T. & Zhang, A. (2013).
    Compressed sensing and affine rank minimization under restricted isometry.
    IEEE Transactions on Signal Processing 61, 3279-3290.

    Cai, T. & Zhang, A. (2013).
    Sharp RIP bound for sparse signal and low-rank matrix recovery.
    Applied and Computational Harmonic Analysis 35, 74-93

 


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