cs.LGSep 9, 2026

Muon-C: Operator-Aligned Muon for Convolutional Kernels

Authors: Jiaxin Qing, Lexin Li

Abstract

Muon replaces matrix momentum with an approximately orthogonal polar direction, but its geometry depends on the matrix representation. For convolution, standard unfolding describes a local patch map rather than the convolution operator. We introduce Muon-C, an operator-aligned optimizer that represents kernel momentum as frequency-wise channel-transfer matrices, polarizes these blocks independently, and uses a critical Fourier grid to return updates exactly to the original finite kernel support. We show that the new geometry arises from combining the block partition and Fourier coordinates. The exact-polar direction is a linear minimization oracle under the critically sampled convolution norm. Its worst-case guarantee relative to the continuous convolution-operator norm is never weaker than unfolding and is strictly stronger for 3×33\times3 kernels. On CIFAR-10 flow matching with matched applied-update RMS, Muon-C reaches 9.87 FID at 40k iterations, compared with 22.26 for unfolded Muon and 51.31 for Adam. It reaches their final quality using 0.62×0.62\times and 0.64×0.64\times their model FLOPs, respectively. Under equal tuning budgets, Muon-C achieves 3.42 FID. Gains persist across data scales and transfer to classification across convolutional architectures.

Explore similar work

CardsList
Jun 11, 2026cs.LG

Muon^p: Muon with Fractional Spectral Powers

Muon is an increasingly widely used optimizer that replaces a gradient G=USV⊤G=USV^\top with its polar factor UV⊤UV^\top, thereby flattening the singular spectrum. However, full flattening discards singular-value information that may matter for adaptation. We introduce Muonp^p, a Muon-style optimizer that instead uses fractional spectral-power updates USpV⊤US^pV^\top for rational p∈(0,1)p\in(0,1), interpolating between Muon and gradient descent. To make it practical, we prove that fractional spectral powers cannot be computed by any fixed univariate polynomial iteration, and furthermore derive low-degree odd bivariate recurrences that approximate USpV⊤US^pV^\top using only matrix multiplications, preserving Muon's matrix-multiplication-only structure and compute complexity. We show that Muonp^p maximizes the linear improvement in loss under the Schatten qq-norm for q=1+1pq=1+\frac{1}{p}. Empirically, Muonp^p is especially effective for finetuning: on billion-scale models, Muonp^p improves validation perplexity and downstream task performance. We further analyze when Muonp^p is less suitable, through the lens of spectral geometry. Our results reveal important insights on when preserving the singular spectrum can bring significant gains, and introduce a principled way to achieve them.
Yihe Dong, Will Sawin
MuonSpectral DescentGradient DescentMatrix Multiplication
AskPDF
Jul 16, 2026cs.LG

Muse: Representation Geometry of Muon Beyond Normalized Momentum

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.
Da Chang, Qiankun Shi, Lvgang Zhang +4
MuonRepresentation SpaceSpectral NormMomentumNormalized MomentumSpectral Descent+1
AskPDF
Jun 15, 2026cs.LG

CacheMuon: Using Temporal Preconditioning To Approximate Polar Factor

Muon is an optimizer that computes updates using the polar factor of the momentum matrix and has shown strong empirical performance across a range of training settings. A key component of Muon is the Newton-Schulz iteration used to compute this polar factor. Although this avoids the cost of an exact singular value decomposition, it remains expensive in practice because it is applied at every optimization step. At the same time, the momentum matrix changes smoothly over training, suggesting strong temporal correlation in the corresponding polar factors. In this paper, we exploit this structure and propose CacheMuon, a temporal preconditioning method that reuses information from previous optimization steps to approximate the polar factor at the current step. This reduces redundant orthogonalization computation across iterations. We analyze CacheMuon as an inexact Muon update, with error controlled by fresh-solver error and cache staleness. Empirically, CacheMuon provides a controllable quality-efficiency frontier: conservative thresholds closely match fresh Muon on language-model and vision training while reducing orthogonalization FLOPs, whereas more aggressive thresholds yield larger arithmetic savings at the cost of modest validation-quality degradation.
Bishnu Dev, Sushil Bohara, Martin Takáč +1
MuonSpectral PreconditioningOrthogonalityNewtonPolarisCache+1
AskPDF