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.

The Information-Theoretic Benefit of Shared Representations under Orthogonality Constraints

Modern deep learning architectures are increasingly multi-task and multi-modal, using a pretrained foundation model combined with task-specific, fine-tuned models. Empirically, exploiting similarity across different problems, instead of solving them individually, can significantly improve overall performance. While the generalization and sample complexity properties of multitask learning have been widely studied, the parametric complexity of joint approximation in comparison to separate approximation remains less well understood. The question is particularly relevant in modern deep learning, where models are increasingly required to satisfy structural constraints such as equivariance, conservation laws, or orthogonality. We prove lower and upper bounds on the description-length for separate and joint approximation classes, respectively, in uniform norm. We build a class of orthogonal functions by composing a shared hard feature, realized by a Rademacher-Haar wavelet series, with Sawtooth-Walsh readouts to enforce orthogonality of output coordinates. The dyadic tree structure of the Rademacher-Haar wavelet concentrates the approximation hardness in the common feature component, while the readouts act as task-specific heads. Using an information-theoretic framework, we obtain a sharp gap between the optimal approximation rates achievable by joint and separate coding. Finally, we realize this separation in a neural network model using Heaviside activations via reduction to triangle-wave approximation. Our results show that even under an orthogonality constraint joint approximation requires strictly fewer bits in compositional architectures, provided the tasks share a latent hard feature. This provides theoretical insight into the description-length-efficiency of compositional multi-output architectures and clarifies how neural networks can retain expressivity under geometric constraints.