Asymptotic Theory for Estimating the Singular Vectors and Values of a Partially-observed Low Rank Matrix with Noise

Matrix completion algorithms recover a low rank matrix from a small fraction of the entries, each entry contaminated with additive errors. In practice, the singular vectors and singular values of the low rank matrix play a pivotal role for statistical analyses and inferences. This paper proposes and studies estimators of these quantities. When the dimensions of the matrix increase to infinity and the probability of observing each entry is identical, Theorem 4.2 gives the asymptotic distribution of the estimated singular values. Even though the estimators use only a partially observed matrix, they achieve the same rates of convergence as the fully observed case. The ensuing estimator of the full low rank matrix is computed with a non-iterative algorithm and it provides a consistent estimator. In the cases studied in this paper, its convergence rate is comparable to the convergence rate of iterative matrix completion algorithms. The numerical experiments corroborate our theoretical results.