{
  "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 $3\\times3$ 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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.09676",
  "author_total": 2,
  "authors": [
    {
      "name": "Jiaxin Qing"
    },
    {
      "name": "Lexin Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8d9be5vpjfv2e.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.09676",
  "fulltext_url": null,
  "id": "8d9be5vpjfv2e",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.09676",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "openalex",
      "1702|Stochastic Gradient Optimization Techniques"
    ]
  ],
  "openalex": "W7212180974",
  "pdf_url": "https://arxiv.org/pdf/2609.09676",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8d9be5vpjfv2e.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:22Z",
      "native_id": "2609.09676",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.09676"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Muon-C: Operator-Aligned Muon for Convolutional Kernels",
  "title_text": "Muon-C: Operator-Aligned Muon for Convolutional Kernels",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:22Z",
  "url": "https://research.draconitic.com/paper/8d9be5vpjfv2e",
  "venue": "arXiv",
  "versions": []
}