# A Riemannian rank-adaptive method for low-rank matrix completion

arXiv: https://arxiv.org/abs/2103.14768 · HTML: https://mlsanity.com/paper/2103.14768/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 26 Mar 2021.

## Built on by (most cited first, 4 of 4)

- [Low-rank lottery tickets: finding efficient low-rank neural networks via matrix differential equations](https://mlsanity.com/paper/2205.13571/) (2022)
- [Nonconvex Factorization and Manifold Formulations are Almost Equivalent in Low-rank Matrix Optimization](https://mlsanity.com/paper/2108.01772/) (2021)
- [A space-decoupling framework for optimization on bounded-rank matrices with orthogonally invariant constraints](https://mlsanity.com/paper/2501.13830/) (2025)
- [Riemannian Gradient Descent for Low-Rank Architectures](https://mlsanity.com/paper/2606.02328/) (2026)

## Abstract (arXiv)

The low-rank matrix completion problem can be solved by Riemannian optimization on a fixed-rank manifold. However, a drawback of the known approaches is that the rank parameter has to be fixed a priori. In this paper, we consider the optimization problem on the set of bounded-rank matrices. We propose a Riemannian rank-adaptive method, which consists of fixed-rank optimization, rank increase step and rank reduction step. We explore its performance applied to the low-rank matrix completion problem. Numerical experiments on synthetic and real-world datasets illustrate that the proposed rank-adaptive method compares favorably with state-of-the-art algorithms. In addition, it shows that one can incorporate each aspect of this rank-adaptive framework separately into existing algorithms for the purpose of improving performance.