{
  "abstract": "The rank $r$ in LoRA is widely treated as a capacity control: a smaller rank is assumed to yield a simpler model that generalizes better. We show that, under hard per-factor norm budgets---the idealization of the weight decay and norm control used in practice---this intuition breaks down. The reason is structural: under such budgets, the updates LoRA can reach are exactly the matrices of rank at most $r$ inside a nuclear-norm ball, and every complexity and displacement functional we analyze is maximized over this set by a rank-one update---so the rank cap never binds. The consequences follow directly. The linear-readout model class we study is identical for every $r \\ge 1$, its Rademacher complexity carries no dependence on $r$, and the distance the adaptation can move the source distribution obeys a rank-independent upper bound that we show is sharp. If rank does not control capacity, where does it act? We identify two places. Statistically, replacing the per-factor budgets with a joint budget on the product restores a data-dependent, rank-sensitive complexity bound---though the gain appears only for well-spread feature distributions, and the worst case remains rank-free. Spectrally, rank sets the price of adaptation: canceling the leading singular directions of the pretrained weight requires both sufficient rank and sufficient budget. We bound the smallest rank achieving a desired source--target alignment, with upper and lower bounds that match under two-sided spectral decay. Together, these results recast rank as governing which updates are reachable and what cancellation costs---not how much capacity the model has.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.32002",
  "author_total": 1,
  "authors": [
    {
      "name": "Babak Barazandeh"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9j0qa45jse160.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.32002",
  "fulltext_url": null,
  "id": "9j0qa45jse160",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.32002",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.AI"
    ],
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "openalex",
      "1702|Domain Adaptation and Few-Shot Learning"
    ]
  ],
  "openalex": "W7214725226",
  "pdf_url": "https://arxiv.org/pdf/2609.32002",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9j0qa45jse160.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:13Z",
      "native_id": "2609.32002",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.32002"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "What Does the Rank Buy? A Spectral and Distributional Analysis of Low-Rank Adaptation",
  "title_text": "What Does the Rank Buy? A Spectral and Distributional Analysis of Low-Rank Adaptation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:13Z",
  "url": "https://research.draconitic.com/paper/9j0qa45jse160",
  "venue": "arXiv",
  "versions": []
}