Muon在矩阵优化里挺火，但上Stiefel流形一直靠近似。这篇论文给了精确闭式解，还做了个Skewon，收敛性有保障，做正交约束优化的值得看。
Muon是一种矩阵感知优化方法，用于Stiefel流形（正交列矩阵集合）上的优化。以往将Muon扩展到该流形的做法依赖启发式或迭代近似。新论文证明Stiefel Muon更新存在精确闭式解，并据此实现高效算法Skewon。Skewon在平滑非凸条件下具有一阶收敛保证。
Muon on the Stiefel Manifold Admits an Exact Closed-Form Update
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.