# Riemannian preconditioning

arXiv: https://arxiv.org/abs/1405.6055 · HTML: https://mlsanity.com/paper/1405.6055/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 May 2014.

## Built on by (most cited first, 19 of 19)

- [Riemannian Preconditioned LoRA for Fine-Tuning Foundation Models](https://mlsanity.com/paper/2402.02347/) (2024)
- [Learning with symmetric positive definite matrices via generalized Bures-Wasserstein geometry](https://mlsanity.com/paper/2110.10464/) (2021)
- [A Stronger Mixture of Low-Rank Experts for Fine-Tuning Foundation Models](https://mlsanity.com/paper/2502.15828/) (2025)
- [AltLoRA: Towards Better Gradient Approximation in Low-Rank Adaptation with Alternating Projections](https://mlsanity.com/paper/2505.12455/) (2025)
- [Manifold constrained steepest descent for smooth and closed-set optimization](https://mlsanity.com/paper/2601.21487/) (2026)
- [Riemannian block SPD coupling manifold and its application to optimal transport](https://mlsanity.com/paper/2201.12933/) (2022)
- [Optimization without Retraction on the Random Generalized Stiefel Manifold](https://mlsanity.com/paper/2405.01702/) (2024)
- [Intrinsic Muon: Spectral Optimization on Riemannian Matrix Manifolds](https://mlsanity.com/paper/2605.09238/) (2026)
- [New Riemannian preconditioned algorithms for tensor completion via polyadic decomposition](https://mlsanity.com/paper/2101.11108/) (2021)
- [Nonconvex Factorization and Manifold Formulations are Almost Equivalent in Low-rank Matrix Optimization](https://mlsanity.com/paper/2108.01772/) (2021)
- [Efficient Over-parameterized Matrix Sensing from Noisy Measurements via Alternating Preconditioned Gradient Descent](https://mlsanity.com/paper/2502.00463/) (2025)
- [On Geometric Connections of Embedded and Quotient Geometries in Riemannian Fixed-rank Matrix Optimization](https://mlsanity.com/paper/2110.12121/) (2021)
- [Faster Randomized Methods for Orthogonality Constrained Problems](https://mlsanity.com/paper/2106.12060/) (2021)
- [Scalable Second-order Riemannian Optimization for K-means Clustering](https://mlsanity.com/paper/2509.21675/) (2025)
- [A Unified Study of LoRA Variants: Taxonomy, Review, Codebase, and Empirical Evaluation](https://mlsanity.com/paper/2601.22708/) (2026)
- [Synthetic Principal Component Design: Fast Covariate Balancing with Synthetic Controls](https://mlsanity.com/paper/2211.15241/) (2022)
- [A Riemannian Approach to Low-Rank Optimal Transport](https://mlsanity.com/paper/2606.12120/) (2026)
- [Riemannian Optimization for Hadamard Products of Low-Rank Matrices](https://mlsanity.com/paper/2606.01216/) (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)

This paper exploits a basic connection between sequential quadratic programming and Riemannian gradient optimization to address the general question of selecting a metric in Riemannian optimization, in particular when the Riemannian structure is sought on a quotient manifold. The proposed method is shown to be particularly insightful and efficient in quadratic optimization with orthogonality and/or rank constraints, which covers most current applications of Riemannian optimization in matrix manifolds.