Muse: Representation Geometry of Muon Beyond Normalized Momentum

• URL: http://arxiv.org/abs/2607.14536v1
• Date: Thu, 16 Jul 2026 03:42:21 GMT
• Title: Muse: Representation Geometry of Muon Beyond Normalized Momentum
• Abstract summary: Muon-styles apply a polar map to matrix momentum, but their updates also depend on the representation of each parameter block before curvatureization.<n>We introduce method, a family of Muon-styles that shares the same momentum rule and Newton's momentum rule.<n>We show that balanced non-native representations can match the native representation, whereas reducing the shorter dimension weakens the scaling and singular-channel support, leading to behavior that increasingly normalized momentum.
• Score: 26.90642116529718
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Muon-style optimizers apply a polar map to matrix momentum, but their updates also depend on the representation of each parameter block before orthogonalization. We study this representation choice as a form of optimizer geometry and introduce {\method}, a family of Muon-style optimizers that shares the same momentum rule and Newton--Schulz backend across native, nearest-square, skinny, and vector representations. Each Frobenius-isometric representation induces a distinct polar steepest-descent geometry, in which the shorter matrix dimension determines the number of supported singular channels, the pullback scaling, and the constants in stochastic nonconvex convergence bounds. In a teacher--student model, curvature collapse and an isotropic Marchenko--Pastur spectral profile connect early-stage dissipation to the represented nuclear-to-squared-Frobenius norm ratio. Pretraining experiments on LLaMA2-130M and LLaMA2-600M, together with fixed-momentum diagnostics, show that balanced non-native representations can match the performance of the native representation, whereas reducing the shorter dimension weakens the scaling and singular-channel support, leading to behavior that increasingly resembles normalized momentum.

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