The Spectral Dynamics and Noise Geometry of Muon

Muon replaces a matrix gradient \(G=U\Sigma V^\top\) by its polar factor \(UV^\top\). This keeps the singular directions selected by the gradient, but makes the update spectrum flat. We study the optimization bias created by this operation. In an underdetermined regression model, we derive exact singular-value dynamics for a projected polar flow and identify a measurement-dependent condition under which the normalized spectrum moves toward equal nonzero singular values. At fixed Frobenius norm, the pairwise spectral functional is minimized by a flat spectrum, whereas nuclear-norm minimization favors spectral concentration. Controlled matrix-sensing experiments separate the effect from simple gradient rescaling, show that norm-matched gradient descent does not reproduce Muon, and recover the predicted flattening trend across broad ablations. In small NanoGPT pretraining, Muon preserves stable rank, has a broad learning-rate plateau, and improves validation loss relative to AdamW; in a matched small-ViT control, the ranking reverses. The resulting picture is regime-dependent: Muon is not universally superior, but its flat-spectrum bias can help when many spectral directions need to remain active.