{
  "abstract": "Transformers have exhibited impressive empirical success across various domains, but their theoretical foundations remain less developed. This work constitutes a mathematical study of the measure-to-measure operators defined by transformers. We show that transformers map sub-Gaussian inputs to sub-Gaussian outputs; this ensures that taking arbitrary-length compositions of the softmax operator is well-defined. We then show that transformers are Hölder continuous with respect to the 1-Wasserstein distance on appropriate spaces of sub-Gaussian inputs. This allows us to establish estimates on the error propagation along a transformer between a sub-Gaussian input and its empirical approximation. We also study a mean-field analog of the cross-attention mechanism, which is an operator from a pair of probability measures to a single probability measure. We show that cross-attention exhibits different Hölder regularity and sample-complexity in its two input arguments. Last, we apply our results to deduce approximation guarantees for measure-to-measure transformers. Together, these results provide a firm stability and finite-sample theory for transformers on sub-Gaussian data.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.07717",
  "author_total": 3,
  "authors": [
    {
      "name": "Frank Cole"
    },
    {
      "name": "Nicholas H. Nelsen"
    },
    {
      "name": "Takashi Furuya"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9t2mec0tjdat7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.07717",
  "fulltext_url": null,
  "id": "9t2mec0tjdat7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.07717",
  "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": "W7220965020",
  "pdf_url": "https://arxiv.org/pdf/2610.07717",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9t2mec0tjdat7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:32Z",
      "native_id": "2610.07717",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.07717"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Stability of Measure-to-Measure Transformers on Sub-Gaussian Data",
  "title_text": "Stability of Measure-to-Measure Transformers on Sub-Gaussian Data",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:32Z",
  "url": "https://research.draconitic.com/paper/9t2mec0tjdat7",
  "venue": "arXiv",
  "versions": []
}