{
  "abstract": "Does representation learning stop when the training loss stops improving? We study this question for matrix Muon, whose polar-normalised updates have a step length set by the gradient's rank rather than its norm. Near the edge of stability, full-batch Muon on teacher-student problems enters approximately period-2 loss oscillations that persist for thousands of steps: the cycle-mean loss stays flat or rises, yet the weights keep moving and the learned features continue to align with the teacher subspace. For linear teacher-student learning toys, we derive explicit cycle and alignment formulas and conditional plateau and decay bounds. For a population mean-field ReLU model, we prove that, under stated dimension, initialisation and small-head conditions, the leading eigenspace of the average gradient outer product (AGOP) recovers the teacher subspace exactly during a loss plateau, before the loss later drops. In all 33 ReLU, GELU and SiLU teacher configurations we study, direction-only alignment metrics show the student AGOP aligned with, or still aligning to, the teacher subspace during the period-2 oscillations; projected head refitting on selected configurations shows that the learned directions are useful for prediction, and further measurements distinguish AGOP alignment from weight-mass concentration. In deep residual ReLU students, freezing the downstream layers while the first layer trains with full-batch exact polar updates recreates a nearly flat cycle-mean loss with improving input-AGOP alignment; freezing and unfreezing switch between this plateau and loss decrease, and the effect is sensitive to momentum and to the choice of orthogonaliser.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.36240",
  "author_total": 1,
  "authors": [
    {
      "name": "Akash Kumar"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5xqq1q6tcxy0x.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.36240",
  "fulltext_url": null,
  "id": "5xqq1q6tcxy0x",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.36240",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "openalex",
      "1702|Stochastic Gradient Optimization Techniques"
    ]
  ],
  "openalex": "W7214889453",
  "pdf_url": "https://arxiv.org/pdf/2609.36240",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5xqq1q6tcxy0x.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:35Z",
      "native_id": "2609.36240",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.36240"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Can Representation Learning Decouple from Loss Minimization? Polar Updates Have an Answer",
  "title_text": "Can Representation Learning Decouple from Loss Minimization? Polar Updates Have an Answer",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:41Z",
  "url": "https://research.draconitic.com/paper/5xqq1q6tcxy0x",
  "venue": "arXiv",
  "versions": []
}