ROP: Matrix recovery via rank-one projections
This addresses matrix recovery for applications like high-dimensional statistics, offering a robust method with proven optimality, though it is incremental as it builds on existing nuclear norm techniques.
The paper tackles low-rank matrix recovery from rank-one projections by proposing a constrained nuclear norm minimization method, achieving rate-optimal estimation with bounds under Frobenius norm loss and demonstrating accurate covariance estimation from one-dimensional projections.
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.