MuCon: Clipped Muon Updates for LLM Training

Muon-style optimizers take a matrix-valued momentum or preconditioned update B = U operatornamediag(σ1,ldots,σr) V^top and replace it with its canonical partial polar factor operatornamePol(B) = U V^top. This maps every nonzero singular value to one. MuCon is the clipped-Muon variant studied here: it applies singular-value clipping to the same Muon matrix, DMuConτ(B) = operatornameMClipτ(B) = U operatornamediagbigl(minσi,τbigr) V^top, qquad τ > 0. Thus, operatornameMClipτ denotes the mathematical clipping operator, while MuCon denotes the optimizer primitive that substitutes this clipped direction for Muon's polar direction. The Muon/MuCon scaling parameterization used in this work is called SpectralP: it is the hidden-matrix scaling recipe under which polar Muon or clipped MuCon directions are applied. The map operatornameMClipτ is the Frobenius projection onto the spectral-norm ball X : |X|2 ≤ τ: it leaves singular values at or below τ unchanged and modifies only the violating singular directions. This paper asks when the MuCon clipping step can be approximated without a full dense SVD. We record two exact identities, a polar/absolute-value formula and a scalar-root formulation leading to a rational Newton filter for the clipped positive-semidefinite factor, and identify the numerical obstruction common to both: singular values near the threshold make sign decisions and rational solves ill-conditioned. Matrix-function methods are therefore useful only when paired with stable polar/square-root primitives or explicit regularization near the clipping boundary.