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

TL;DR - An arXiv preprint showing that the Muon matrix-aware optimizer's update on the Stiefel manifold (matrices with orthonormal columns) has an exact closed-form solution, yielding a new algorithm called Skewon. It matters because prior Stiefel extensions of Muon relied on heuristic, approximate, or iterative updates with uneven efficiency.

• Derives an exact closed-form solution for the Stiefel Muon update, replacing approximate/iterative schemes used in existing extensions.
• Introduces Skewon, a practical algorithm for orthogonality-constrained optimization with an efficient implementation built on that closed form.
• Establishes first-order convergence guarantees for Skewon in the smooth non-convex setting.
• Targets a constraint set (orthonormal columns) that is common in machine learning and scientific computing; abstract reports no empirical benchmarks, so speed/quality gains are unquantified here.