# Inference and Uncertainty Quantification for Noisy Matrix Completion

arXiv: https://arxiv.org/abs/1906.04159 · HTML: https://mlsanity.com/paper/1906.04159/

## Key facts

- Cited by 39 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Jun 2019.

## Built on by (most cited first, 30 of 39)

- [Causal Matrix Completion](https://mlsanity.com/paper/2109.15154/) (2021)
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- [Inference for Heteroskedastic PCA with Missing Data](https://mlsanity.com/paper/2107.12365/) (2021)
- [Provable Tensor-Train Format Tensor Completion by Riemannian Optimization](https://mlsanity.com/paper/2108.12163/) (2021)
- [Multiple Imputation via Generative Adversarial Network for High-dimensional Blockwise Missing Value Problems](https://mlsanity.com/paper/2112.11507/) (2021)
- [Counterfactual inference in sequential experiments](https://mlsanity.com/paper/2202.06891/) (2022)
- [Improved Global Guarantees for the Nonconvex Burer--Monteiro Factorization via Rank Overparameterization](https://mlsanity.com/paper/2207.01789/) (2022)
- [Model-Based Reinforcement Learning for Offline Zero-Sum Markov Games](https://mlsanity.com/paper/2206.04044/) (2022)
- [Low-Rank Contextual Reinforcement Learning from Heterogeneous Human Feedback](https://mlsanity.com/paper/2412.19436/) (2024)
- [Doubly robust nearest neighbors in factor models](https://mlsanity.com/paper/2211.14297/) (2022)
- [Deflated HeteroPCA: Overcoming the curse of ill-conditioning in heteroskedastic PCA](https://mlsanity.com/paper/2303.06198/) (2023)
- [Concentration properties of fractional posterior in 1-bit matrix completion](https://mlsanity.com/paper/2404.08969/) (2024)
- [Learning Treatment Effects in Panels with General Intervention Patterns](https://mlsanity.com/paper/2106.02780/) (2021)
- [Inference of Dependency Knowledge Graph for Electronic Health Records](https://mlsanity.com/paper/2312.15611/) (2023)
- [Robust Matrix Completion with Heavy-tailed Noise](https://mlsanity.com/paper/2206.04276/) (2022)
- [Multi-User Reinforcement Learning with Low Rank Rewards](https://mlsanity.com/paper/2210.05355/) (2022)
- [PRISM: Distribution-free Adaptive Computation of Matrix Functions for Accelerating Neural Network Training](https://mlsanity.com/paper/2601.22137/) (2026)
- [Online Policy Learning and Inference by Matrix Completion](https://mlsanity.com/paper/2404.17398/) (2024)
- [LEARNER: A Transfer Learning Method for Low-Rank Matrix Estimation](https://mlsanity.com/paper/2412.20605/) (2024)
- [Multi-dimensional domain generalization with low-rank structures](https://mlsanity.com/paper/2309.09555/) (2023)
- [Statistical Inference For Noisy Matrix Completion Incorporating Auxiliary Information](https://mlsanity.com/paper/2403.14899/) (2024)
- [Uncertainty Quantification For Low-Rank Matrix Completion With Heterogeneous and Sub-Exponential Noise](https://mlsanity.com/paper/2110.12046/) (2021)
- [Using Interpretable Machine Learning to Massively Increase the Number of Antibody-Virus Interactions Across Studies](https://mlsanity.com/paper/2206.14566/) (2022)
- [A Generalized Latent Factor Model Approach to Mixed-data Matrix Completion with Entrywise Consistency](https://mlsanity.com/paper/2211.09272/) (2022)
- [Inferences on Mixing Probabilities and Ranking in Mixed-Membership Models](https://mlsanity.com/paper/2308.14988/) (2023)
- [Statistical Inference in Tensor Completion: Optimal Uncertainty Quantification and Statistical-to-Computational Gaps](https://mlsanity.com/paper/2410.11225/) (2024)
- [Time-Series Imputation with Wasserstein Interpolation for Optimal Look-Ahead-Bias and Variance Tradeoff](https://mlsanity.com/paper/2102.12736/) (2021)
- [Simulation comparisons between Bayesian and de-biased estimators in low-rank matrix completion](https://mlsanity.com/paper/2103.11749/) (2021)
- [PAC-Bayesian Matrix Completion with a Spectral Scaled Student Prior](https://mlsanity.com/paper/2104.08191/) (2021)

## Abstract (arXiv)

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 citechen2019noisy.