# Muon on the Stiefel Manifold Admits an Exact Closed-Form Update

arXiv: https://arxiv.org/abs/2608.06218 · HTML: https://mlsanity.com/paper/2608.06218/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 21 references that are in the map.
- First posted on arXiv 6 Aug 2026.

## Builds on (most cited first, 8 of 8)

- [The Polar Express: Optimal Matrix Sign Methods and Their Application to the Muon Algorithm](https://mlsanity.com/paper/2505.16932/) (2025)
- [PoLAR: Polar-Decomposed Low-Rank Adapter Representation](https://mlsanity.com/paper/2506.03133/) (2025)
- [LoRA meets Riemannion: Muon Optimizer for Parametrization-independent Low-Rank Adapters](https://mlsanity.com/paper/2507.12142/) (2025)
- [Insights on Muon from Simple Quadratics](https://mlsanity.com/paper/2602.11948/) (2026)
- [Manifold constrained steepest descent for smooth and closed-set optimization](https://mlsanity.com/paper/2601.21487/) (2026)
- [Intrinsic Muon: Spectral Optimization on Riemannian Matrix Manifolds](https://mlsanity.com/paper/2605.09238/) (2026)
- [OrthoFuse: Training-free Riemannian Fusion of Orthogonal Style-Concept Adapters for Diffusion Models](https://mlsanity.com/paper/2604.05183/) (2026)
- [Tensorion: A Tensor-Aware Generalization of the Muon Optimizer](https://mlsanity.com/paper/2606.25975/) (2026)

## Abstract (arXiv)

We study Muon, a recently proposed matrix-aware optimization method, in the context of the Stiefel manifold. This manifold consists of matrices with orthonormal columns and is ubiquitous in machine learning and scientific computing. Existing extensions of Muon to this manifold rely on heuristic, approximate, or iterative updates with varying computational efficiency. We show that the corresponding Stiefel Muon update admits an exact closed-form solution and use this result to develop Skewon, a practical algorithm for orthogonality-constrained optimization with an efficient implementation. We further establish first-order convergence guarantees for Skewon in the smooth non-convex setting.