# Inference and Uncertainty Quantification for Noisy Matrix Completion

## TL;DR

This paper introduces a bias correction method for noisy matrix completion estimators, enabling precise statistical inference and confidence interval construction without data splitting, and achieving optimal efficiency.

## Contribution

It develops a de-biased estimator for noisy matrix completion that allows for valid uncertainty quantification and confidence intervals, advancing the statistical inference capabilities in this area.

## Key findings

- De-biased estimators admit nearly precise non-asymptotic distributional characterizations.
- Constructs valid confidence intervals for missing entries and low-rank factors.
- Achieves full statistical efficiency, including the preconstant, for the estimators.

## Abstract

Noisy matrix completion aims at estimating a low-rank matrix given only partial and corrupted entries. Despite substantial progress in designing efficient estimation algorithms, it remains largely unclear how to assess the uncertainty of the obtained estimates and how to perform statistical inference on the unknown matrix (e.g.~constructing a valid and short confidence interval for an unseen entry). This paper takes a step towards inference and uncertainty quantification for noisy matrix completion. We develop a simple procedure to compensate for the bias of the widely used convex and nonconvex estimators. The resulting de-biased estimators admit nearly precise non-asymptotic distributional characterizations, which in turn enable optimal construction of confidence intervals\,/\,regions for, say, the missing entries and the low-rank factors. Our inferential procedures do not rely on sample splitting, thus avoiding unnecessary loss of data efficiency. As a byproduct, we obtain a sharp characterization of the estimation accuracy of our de-biased estimators, which, to the best of our knowledge, are the first tractable algorithms that provably achieve full statistical efficiency (including the preconstant). The analysis herein is built upon the intimate link between convex and nonconvex optimization --- an appealing feature recently discovered by \cite{chen2019noisy}.