# A Preconditioned Riemannian Gradient Descent Algorithm for Low-Rank Matrix Recovery

arXiv: https://arxiv.org/abs/2305.02543 · HTML: https://mlsanity.com/paper/2305.02543/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 May 2023.

## Built on by (most cited first, 7 of 7)

- [Low-rank Orthogonalization for Large-scale Matrix Optimization with Applications to Foundation Model Training](https://mlsanity.com/paper/2509.11983/) (2025)
- [Manifold constrained steepest descent for smooth and closed-set optimization](https://mlsanity.com/paper/2601.21487/) (2026)
- [Efficient Over-parameterized Matrix Sensing from Noisy Measurements via Alternating Preconditioned Gradient Descent](https://mlsanity.com/paper/2502.00463/) (2025)
- [Randomized Approach to Matrix Completion: Applications in Recommendation Systems and Image Inpainting](https://mlsanity.com/paper/2403.01919/) (2024)
- [Fast and Provable Tensor-Train Format Tensor Completion via Precondtioned Riemannian Gradient Descent](https://mlsanity.com/paper/2501.13385/) (2025)
- [AdaPreLoRA: Adafactor Preconditioned Low-Rank Adaptation](https://mlsanity.com/paper/2605.08734/) (2026)
- [Guaranteed Low-Rank Tensor Recovery from Modewise Measurements via Normalized Block-Weighted Riemannian Gradient Descent](https://mlsanity.com/paper/2609.24679/) (2026)

## Abstract (arXiv)

The low-rank matrix recovery problem often arises in various fields, including signal processing, machine learning, and imaging science. The Riemannian gradient descent (RGD) algorithm has proven to be an efficient algorithm for solving this problem. In this paper, we present a preconditioned Riemannian gradient descent (PRGD) for low-rank matrix recovery. The preconditioner, noted for its simplicity and computational efficiency, is constructed by weighting the (i,j)-th entry of the gradient matrix according to the norms of the i-th row and the j-th column. We establish the theoretical recovery guarantee for PRGD under the restricted isometry property assumption. Experimental results indicate that PRGD can accelerate RGD by up to tenfold in solving low-rank matrix recovery problems such as matrix completion.