  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.
