Matrix completion algorithms recover a low rank matrix from a small fractionof the entries, each entry contaminated with additive errors. In practice, thesingular vectors and singular values of the low rank matrix play a pivotal rolefor statistical analyses and inferences. This paper proposes estimators ofthese quantities and studies their asymptotic behavior. Under the setting wherethe dimensions of the matrix increase to infinity and the probability ofobserving each entry is identical, Theorem 4.1 gives the rate of convergencefor the estimated singular vectors; Theorem 4.3 gives a multivariate centrallimit theorem for the estimated singular values. Even though the estimators useonly a partially observed matrix, they achieve the same rates of convergence asthe fully observed case. These estimators combine to form a consistentestimator of the full low rank matrix that is computed with a non-iterativealgorithm. In the cases studied in this paper, this estimator achieves theminimax lower bound in Koltchinskii et al. (2011). The numerical experimentscorroborate our theoretical results.
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