Asymptotic theory for estimating the singular vectors and values of a partially-observed low rank matrix with noise

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 et al. (2011a). The numerical experiments corroborate our theoretical results.

Rate-optimal perturbation bounds for singular subspaces with applications to high-dimensional statistics

“…Matrix completion, whose central goal is to recover a large low-rank matrix based on a limited number of observable entries, has been widely studied in the last decade. Among various methods for matrix completion, spectral method is fast, easy to implement and achieves good performance (Keshavan et al, 2010;Chatterjee, 2014;Cho et al, 2015). The new perturbation bounds can be potentially used for singular space estimation under the matrix completion setting to yield better results.…”

Intelligent Initialization and Adaptive Thresholding for Iterative Matrix Completion: Some Statistical and Algorithmic Theory forAdaptive-Impute

“…This means that a non-vanishing proportion of entries of M 0 contains non-vanishing signals with dimensionality (see Fan et al (2013)). For more discussion, see Remark 2 in Cho et al (2016).…”

“…Lemma 2 in Cho et al (2016) provides justification of using the scree plot and the singular value gap to choose the rank. For the thresholding parameters for softImpute-type algorithms, we chose the optimal values which result in the smallest test errors.…”

Artificial intelligence enhanced two-dimensional nanoscale nuclear magnetic resonance spectroscopy

“…Inspired by unobserved entries fill strategy 21,22 and matrix completion based domain adaptation [23][24][25] , we propose to combine the deep learning network with the matrix completion algorithm 15 to develop our DLMC method for efficient NV spectrum map reconstruction. DL can learn very complex non-linear mapping from a partially filled spectrum map to its full-resolution map, with the DL network trained with simulation data; while the traditional matrix completion (MC) method is used for post-processing the DL output map to keep its low-rank property, thus further alleviate the domain shift problem.…”