# Optimization flows landing on the Stiefel manifold

arXiv: https://arxiv.org/abs/2202.09058 · HTML: https://mlsanity.com/paper/2202.09058/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Feb 2022.

## Built on by (most cited first, 10 of 10)

- [PoLAR: Polar-Decomposed Low-Rank Adapter Representation](https://mlsanity.com/paper/2506.03133/) (2025)
- [Infeasible Deterministic, Stochastic, and Variance-Reduction Algorithms for Optimization under Orthogonality Constraints](https://mlsanity.com/paper/2303.16510/) (2023)
- [Orthogonal Directions Constrained Gradient Method: from non-linear equality constraints to Stiefel manifold](https://mlsanity.com/paper/2303.09261/) (2023)
- [Optimization without Retraction on the Random Generalized Stiefel Manifold](https://mlsanity.com/paper/2405.01702/) (2024)
- [Stiefel Flow Matching for Moment-Constrained Structure Elucidation](https://mlsanity.com/paper/2412.12540/) (2024)
- [Low-Rank Adaptation Redux for Large Models](https://mlsanity.com/paper/2604.21905/) (2026)
- [Bounds on the geodesic distances on the Stiefel manifold for a family of Riemannian metrics](https://mlsanity.com/paper/2408.07072/) (2024)
- [Scalable Variational Bayesian Fine-Tuning of LLMs via Orthogonalized Low-Rank Adapters](https://mlsanity.com/paper/2604.03388/) (2026)
- [A second-order method landing on the Stiefel manifold via Newtonunicodex2013Schulz iteration](https://mlsanity.com/paper/2605.02838/) (2026)
- [Optimization over the intersection of manifolds](https://mlsanity.com/paper/2605.22736/) (2026)

## Abstract (arXiv)

We study a continuous-time system that solves optimization problems over the set of orthonormal matrices, which is also known as the Stiefel manifold. The resulting optimization flow follows a path that is not always on the manifold but asymptotically lands on the manifold. We introduce a generalized Stiefel manifold to which we extend the canonical metric of the Stiefel manifold. We show that the vector field of the proposed flow can be interpreted as the sum of a Riemannian gradient on a generalized Stiefel manifold and a normal vector. Moreover, we prove that the proposed flow globally converges to the set of critical points, and any local minimum and isolated critical point is asymptotically stable.