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

arXiv: https://arxiv.org/abs/1508.05431 · HTML: https://mlsanity.com/paper/1508.05431/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Aug 2015.

## Built on by (most cited first, 3 of 3)

- [Inference for Heteroskedastic PCA with Missing Data](https://mlsanity.com/paper/2107.12365/) (2021)
- [Dimension Reduction for Data with Heterogeneous Missingness](https://mlsanity.com/paper/2109.11765/) (2021)
- [Integrated Analysis for Electronic Health Records with Structured and Sporadic Missingness](https://mlsanity.com/paper/2506.09208/) (2025)

## Abstract (arXiv)

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.