An Isotropy-Preserving Spectral Cap for Muon: Theory and Three Case Studies
Muon and related matrix-sign optimizers are increasingly used to pre-train large language models, but their effect on the internal geometry of individual weight matrices is not well understood. This preliminary report proposes a unified framework built on a single idealizing assumption -- exact scale invariance of the loss under weight rescaling, which holds approximately in normalization-heavy networks. Under this assumption, plain SGD carries a built-in 1/||W|| brake on its update size, whereas Muon's matrix-sign step removes that brake, so both the Frobenius and spectral norms drift outward faster (t^{1/2} versus t^{1/4}). We further observe that the spectral-norm perturbation has a no...
| Authors | Jiachun Li |
|---|---|
| Primary category | cs.LG |
| Published | 22 July 2026 |
| Last updated at source | 22 July 2026 |
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