Muon's Stiefel-Manifold Update Has an Exact Closed Form — "Skewon" Replaces Existing Iterative Approximations

arXiv 2608.06218·low signalXBlueskyLinkedInCopy link

Submitted 2026-08-06 (2608.06218), this paper studies the Muon matrix-aware optimizer on the Stiefel manifold (matrices with orthonormal columns) and shows the corresponding Stiefel Muon update admits an exact closed-form solution, where existing extensions rely on heuristic, approximate, or iterative updates of varying cost. The result becomes Skewon, a practical algorithm for orthogonality-constrained optimization with an efficient implementation, plus first-order convergence guarantees in the smooth non-convex setting. The abstract reports no empirical speedups or benchmarks, so the contribution is theoretical — but Muon's rapid adoption in large-model pretraining makes the closed form immediately relevant to anyone maintaining an optimizer implementation.