# Fast Exact Matrix Completion with Finite Samples

arXiv: https://arxiv.org/abs/1411.1087 · HTML: https://mlsanity.com/paper/1411.1087/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Nov 2014.

## Built on by (most cited first, 15 of 15)

- [Low Rank Matrix Completion via Robust Alternating Minimization in Nearly Linear Time](https://mlsanity.com/paper/2302.11068/) (2023)
- [Matrix Completion with Cross-Concentrated Sampling: Bridging Uniform Sampling and CUR Sampling](https://mlsanity.com/paper/2208.09723/) (2022)
- [GNMR: A provable one-line algorithm for low rank matrix recovery](https://mlsanity.com/paper/2106.12933/) (2021)
- [Private Alternating Least Squares: Practical Private Matrix Completion with Tighter Rates](https://mlsanity.com/paper/2107.09802/) (2021)
- [Accelerating SGD for Highly Ill-Conditioned Huge-Scale Online Matrix Completion](https://mlsanity.com/paper/2208.11246/) (2022)
- [Matrix Completion in Almost-Verification Time](https://mlsanity.com/paper/2308.03661/) (2023)
- [Exact Linear Convergence Rate Analysis for Low-Rank Symmetric Matrix Completion via Gradient Descent](https://mlsanity.com/paper/2102.02396/) (2021)
- [Online Learning and Information Exponents: On The Importance of Batch size, and Time/Complexity Tradeoffs](https://mlsanity.com/paper/2406.02157/) (2024)
- [On Asymptotic Linear Convergence of Projected Gradient Descent for Constrained Least Squares](https://mlsanity.com/paper/2112.11760/) (2021)
- [Matrix Completion with Heterogonous Cost](https://mlsanity.com/paper/2203.12120/) (2022)
- [Leave-One-Out Analysis for Nonconvex Robust Matrix Completion with General Thresholding Functions](https://mlsanity.com/paper/2407.19446/) (2024)
- [Randomized Approach to Matrix Completion: Applications in Recommendation Systems and Image Inpainting](https://mlsanity.com/paper/2403.01919/) (2024)
- [Efficient Federated Low Rank Matrix Completion](https://mlsanity.com/paper/2405.06569/) (2024)
- [Predictive Low Rank Matrix Learning under Partial Observations: Mixed-Projection ADMM](https://mlsanity.com/paper/2407.13731/) (2024)
- [Efficient Tensor Completion Algorithms for Highly Oscillatory Operators](https://mlsanity.com/paper/2510.17734/) (2025)

## Abstract (arXiv)

Matrix completion is the problem of recovering a low rank matrix by observing a small fraction of its entries. A series of recent works [KOM12,JNS13,HW14] have proposed fast non-convex optimization based iterative algorithms to solve this problem. However, the sample complexity in all these results is sub-optimal in its dependence on the rank, condition number and the desired accuracy. In this paper, we present a fast iterative algorithm that solves the matrix completion problem by observing O(nr5 log3 n) entries, which is independent of the condition number and the desired accuracy. The run time of our algorithm is O(nr7log3 nlog 1/ε) which is near linear in the dimension of the matrix. To the best of our knowledge, this is the first near linear time algorithm for exact matrix completion with finite sample complexity (i.e. independent of ε). Our algorithm is based on a well known projected gradient descent method, where the projection is onto the (non-convex) set of low rank matrices. There are two key ideas in our result: 1) our argument is based on a ℓ∞ norm potential function (as opposed to the spectral norm) and provides a novel way to obtain perturbation bounds for it. 2) we prove and use a natural extension of the Davis-Kahan theorem to obtain perturbation bounds on the best low rank approximation of matrices with good eigen-gap. Both of these ideas may be of independent interest.