# Computational Limits for Matrix Completion

arXiv: https://arxiv.org/abs/1402.2331 · HTML: https://mlsanity.com/paper/1402.2331/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Feb 2014.

## Built on by (most cited first, 9 of 9)

- [Low Rank Matrix Completion via Robust Alternating Minimization in Nearly Linear Time](https://mlsanity.com/paper/2302.11068/) (2023)
- [Sample Efficient Linear Meta-Learning by Alternating Minimization](https://mlsanity.com/paper/2105.08306/) (2021)
- [Private Alternating Least Squares: Practical Private Matrix Completion with Tighter Rates](https://mlsanity.com/paper/2107.09802/) (2021)
- [Deep Linear Networks for Matrix Completion -- An Infinite Depth Limit](https://mlsanity.com/paper/2210.12497/) (2022)
- [Hardness of Maximum Likelihood Learning of DPPs](https://mlsanity.com/paper/2205.12377/) (2022)
- [Entry-Specific Bounds for Low-Rank Matrix Completion under Highly Non-Uniform Sampling](https://mlsanity.com/paper/2403.00184/) (2024)
- [Low-Rank Approximation, Adaptation, and Other Tales](https://mlsanity.com/paper/2408.05883/) (2024)
- [Entry-Specific Matrix Estimation under Arbitrary Sampling Patterns through the Lens of Network Flows](https://mlsanity.com/paper/2409.03980/) (2024)
- [New Hardness Results for Low-Rank Matrix Completion](https://mlsanity.com/paper/2506.18440/) (2025)

## Abstract (arXiv)

Matrix Completion is the problem of recovering an unknown real-valued low-rank matrix from a subsample of its entries. Important recent results show that the problem can be solved efficiently under the assumption that the unknown matrix is incoherent and the subsample is drawn uniformly at random. Are these assumptions necessary? It is well known that Matrix Completion in its full generality is NP-hard. However, little is known if make additional assumptions such as incoherence and permit the algorithm to output a matrix of slightly higher rank. In this paper we prove that Matrix Completion remains computationally intractable even if the unknown matrix has rank 4 but we are allowed to output any constant rank matrix, and even if additionally we assume that the unknown matrix is incoherent and are shown 90% of the entries. This result relies on the conjectured hardness of the 4-Coloring problem. We also consider the positive semidefinite Matrix Completion problem. Here we show a similar hardness result under the standard assumption that Pne NP. Our results greatly narrow the gap between existing feasibility results and computational lower bounds. In particular, we believe that our results give the first complexity-theoretic justification for why distributional assumptions are needed beyond the incoherence assumption in order to obtain positive results. On the technical side, we contribute several new ideas on how to encode hard combinatorial problems in low-rank optimization problems. We hope that these techniques will be helpful in further understanding the computational limits of Matrix Completion and related problems.