Muon-C: Operator-Aligned Muon for Convolutional Kernels

arXiv:2609.09676v1 Announce Type: new 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 3imes3 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 0.62imes and 0.64imes 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.

Related

• PGD-NO: A Neural Operator with Precomputed Geometry Decomposition for 3D Million-scale Physics Simulations
• HAMNO: A Hierarchical Adaptive Multi-scale Neural Operator with Physics-Informed Learning for Dynamical Systems
• The Kuramoto Neural Operator: Learning to Solve PDEs via Coupled Oscillator Dynamics
• A Physics-Informed Hybrid Neural Operator for Transient Magnetization Prediction in Power Magnetics

Source: arXiv cs.LG | 2026-09-10