# R3MC: A Riemannian three-factor algorithm for low-rank matrix completion

arXiv: https://arxiv.org/abs/1306.2672 · HTML: https://mlsanity.com/paper/1306.2672/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Jun 2013.

## Built on by (most cited first, 11 of 11)

- [StelLA: Subspace Learning in Low-rank Adaptation using Stiefel Manifold](https://mlsanity.com/paper/2510.01938/) (2025)
- [A Scalable Second Order Method for Ill-Conditioned Matrix Completion from Few Samples](https://mlsanity.com/paper/2106.02119/) (2021)
- [GNMR: A provable one-line algorithm for low rank matrix recovery](https://mlsanity.com/paper/2106.12933/) (2021)
- [Accelerated Riemannian Optimization: Handling Constraints with a Prox to Bound Geometric Penalties](https://mlsanity.com/paper/2211.14645/) (2022)
- [Riemannian Federated Learning via Averaging Gradient Streams](https://mlsanity.com/paper/2409.07223/) (2024)
- [Get the Best of Both Worlds: Improving Accuracy and Transferability by Grassmann Class Representation](https://mlsanity.com/paper/2308.01547/) (2023)
- [Faster Randomized Methods for Orthogonality Constrained Problems](https://mlsanity.com/paper/2106.12060/) (2021)
- [Streamlining in the Riemannian Realm: Efficient Riemannian Optimization with Loopless Variance Reduction](https://mlsanity.com/paper/2403.06677/) (2024)
- [Accurate and fast matrix factorization for low-rank learning](https://mlsanity.com/paper/2104.10785/) (2021)
- [Riemannian Optimization for Hadamard Products of Low-Rank Matrices](https://mlsanity.com/paper/2606.01216/) (2026)
- [Retraction-Based Gradient Projection Algorithms on Manifolds](https://mlsanity.com/paper/2609.30885/) (2026)

## Abstract (arXiv)

We exploit the versatile framework of Riemannian optimization on quotient manifolds to develop R3MC, a nonlinear conjugate-gradient method for low-rank matrix completion. The underlying search space of fixed-rank matrices is endowed with a novel Riemannian metric that is tailored to the least-squares cost. Numerical comparisons suggest that R3MC robustly outperforms state-of-the-art algorithms across different problem instances, especially those that combine scarcely sampled and ill-conditioned data.