Denoise First, Orthogonalize Later: Understanding Momentum in Muon via Spectral Filtering
Muon has recently demonstrated strong empirical performance in large language model training, but the theoretical role of momentum in Muon remains unclear. Existing analyses of Muon either remove momentum to study spectral updates in isolation, or retain momentum without explaining why it improves empirical performance. Our work bridges this gap by showing momentum in Muon acts as a spectral filter. Under a structured signal-plus-perturbation gradient model, we prove that momentum suppresses perturbations while preserving the dominant signal, thereby enlarging the spectral gap between them. This enlarged gap stabilizes the singular subspaces of the matrix passed to Muon's orthogonalizatio...
| Authors | Xianliang Li, Zihan Zhang, Weiyang Liu et al. |
|---|---|
| Primary category | cs.LG |
| Published | 2 June 2026 |
| Last updated at source | 3 June 2026 |
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