{
  "abstract": "We study the symmetric and antisymmetric parts of bilinear forms in the attention heads of trained large language models. We introduce an orthogonally invariant profile map from real bilinear forms to a three-dimensional simplex and observe that profiles of trained bilinear forms accumulate near profiles of rank-one bilinear forms. We prove that the symmetric part of a bilinear form in an attention head is the sum of a hyperbolic form and a zero form for a Zariski-dense subset of query-key matrices.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.22990",
  "author_total": 1,
  "authors": [
    {
      "name": "Andrew O'Desky"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/df3d7h5yvt62t.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.22990",
  "fulltext_url": null,
  "id": "df3d7h5yvt62t",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.22990",
  "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.SP"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "openalex",
      "1702|Topic Modeling"
    ]
  ],
  "openalex": "W7214023439",
  "pdf_url": "https://arxiv.org/pdf/2609.22990",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/df3d7h5yvt62t.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.22990",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.22990"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.SP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "On attention heads and bilinear forms",
  "title_text": "On attention heads and bilinear forms",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:33Z",
  "url": "https://research.draconitic.com/paper/df3d7h5yvt62t",
  "venue": "arXiv",
  "versions": []
}