Statistica Sinica 27 (2017), 1921-1948

ASYMPTOTIC THEORY FOR ESTIMATING THE

SINGULAR VECTORS AND VALUES OF A PARTIALLY-

OBSERVED LOW RANK MATRIX WITH NOISE

Juhee Cho, Donggyu Kim and Karl Rohe

Fred Hutchinson Cancer Research Center, Princeton University

and University of Wisconsin-Madison

Abstract: 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 1 gives the rate of convergence for the estimated singular vectors; Theorem 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, Lounici and Tsybakov (2011). The numerical experiments corroborate our theoretical results.

Key words and phrases: Low rank matrices, matrix completion, matrix estimation,singular value decomposition.