发表机构
University of California, Berkeley（加州大学伯克利分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
提出算子对齐的Muon-C优化器，通过傅里叶域块极化卷积核动量，在CIFAR-10流匹配中显著提升FID并降低计算开销，且泛化至分类任务。
AI 中文摘要
Muon优化器用近似正交的极方向替代矩阵动量，但其几何性质依赖于矩阵表示。对于卷积，标准展开描述的是局部块映射而非卷积算子本身。我们提出Muon-C，一种算子对齐的优化器，它将核动量表示为按频率划分的通道转移矩阵，独立地对这些块进行极化，并使用关键的傅里叶网格将更新精确地返回到原始有限核支撑上。我们证明新几何性质源于块划分与傅里叶坐标的结合。精确极方向是临界采样卷积范数下的线性最小化预言机。其相对于连续卷积算子范数的最坏情况保证不弱于展开方法，且对于3×3核严格更强。在CIFAR-10流匹配中，在匹配的应用更新RMS下，Muon-C在40k迭代时达到9.87的FID，而展开Muon为22.26，Adam为51.31。它分别使用其模型FLOPs的0.62倍和0.64倍达到它们的最终质量。在相等的调参预算下，Muon-C达到3.42的FID。增益在不同数据规模下持续存在，并迁移到跨卷积架构的分类任务中。
英文摘要
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 $3\times3$ 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 $0.62\times$ and $0.64\times$ 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.
