2015/08/21 by Juhee Cho, Cho, Juhee, Donggyu Kim +3 · 2 citations
Computer Science · Engineering · Mathematics · #FOS: Computer and information sciences #Methodology (stat.ME) #Random Matrices and Applications #Sparse and Compressive Sensing Techniques #Stochastic Gradient Optimization Techniques
paper · pdf · doi:10.48550/arxiv.1508.05431
openalex publication_date 2015/08/21 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
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 estimators of these quantities and studies their asymptotic behavior. Under the setting where the dimensions of the matrix increase to infinity and the probability of observing each entry is identical, Theorem 4.1 gives the rate of convergence for the estimated singular vectors; Theorem 4.3 gives a multivariate central limit theorem for 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. These estimators combine to form a consistent estimator of the full low rank matrix that is computed with a non-iterative algorithm. In the cases studied in this paper, this estimator achieves the minimax lower bound in Koltchinskii et al. (2011). The numerical experiments corroborate our theoretical results.