# Nearly-optimal Robust Matrix Completion

arXiv: https://arxiv.org/abs/1606.07315 · HTML: https://mlsanity.com/paper/1606.07315/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Jun 2016.

## Built on by (most cited first, 13 of 13)

- [Learned Robust PCA: A Scalable Deep Unfolding Approach for High-Dimensional Outlier Detection](https://mlsanity.com/paper/2110.05649/) (2021)
- [Low Rank Matrix Completion via Robust Alternating Minimization in Nearly Linear Time](https://mlsanity.com/paper/2302.11068/) (2023)
- [GNMR: A provable one-line algorithm for low rank matrix recovery](https://mlsanity.com/paper/2106.12933/) (2021)
- [Distribution-Free Matrix Prediction Under Arbitrary Missing Pattern](https://mlsanity.com/paper/2305.11640/) (2023)
- [Matrix Completion in Almost-Verification Time](https://mlsanity.com/paper/2308.03661/) (2023)
- [Accelerating Ill-conditioned Hankel Matrix Recovery via Structured Newton-like Descent](https://mlsanity.com/paper/2406.07409/) (2024)
- [Leave-One-Out Analysis for Nonconvex Robust Matrix Completion with General Thresholding Functions](https://mlsanity.com/paper/2407.19446/) (2024)
- [Non-Convex Optimizations for Machine Learning with Theoretical Guarantee: Robust Matrix Completion and Neural Network Learning](https://mlsanity.com/paper/2306.16557/) (2023)
- [Efficient Federated Low Rank Matrix Completion](https://mlsanity.com/paper/2405.06569/) (2024)
- [AltGDmin: Alternating GD and Minimization for Partly-Decoupled (Federated) Optimization](https://mlsanity.com/paper/2504.14741/) (2025)
- [RGNMR: A Gauss-Newton method for robust matrix completion with theoretical guarantees](https://mlsanity.com/paper/2505.12919/) (2025)
- [Transformed ℓ1 Regularizations for Robust Principal Component Analysis: Toward a Fine-Grained Understanding](https://mlsanity.com/paper/2510.03624/) (2025)
- [Robust Low-Tubal-Rank Tensor Completion under Cross-Concentrated Sampling](https://mlsanity.com/paper/2608.03928/) (2026)

## Abstract (arXiv)

In this paper, we consider the problem of Robust Matrix Completion (RMC) where the goal is to recover a low-rank matrix by observing a small number of its entries out of which a few can be arbitrarily corrupted. We propose a simple projected gradient descent method to estimate the low-rank matrix that alternately performs a projected gradient descent step and cleans up a few of the corrupted entries using hard-thresholding. Our algorithm solves RMC using nearly optimal number of observations as well as nearly optimal number of corruptions. Our result also implies significant improvement over the existing time complexity bounds for the low-rank matrix completion problem. Finally, an application of our result to the robust PCA problem (low-rank+sparse matrix separation) leads to nearly linear time (in matrix dimensions) algorithm for the same; existing state-of-the-art methods require quadratic time. Our empirical results corroborate our theoretical results and show that even for moderate sized problems, our method for robust PCA is an an order of magnitude faster than the existing methods.