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arXiv · 2607.14536

Muse: Representation Geometry of Muon Beyond Normalized Momentum

Da Chang·Qiankun Shi·Lvgang Zhang·Di He·Yaoshuai Ma·Ganzhao Yuan·Yongxiang Liu
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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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Da Chang, Qiankun Shi, Lvgang Zhang, Di He, Yaoshuai Ma, Ganzhao Yuan, Yongxiang Liu. 2026-07-16. Muse: Representation Geometry of Muon Beyond Normalized Momentum. https://arxiv.org/abs/2607.14536

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