{
  "abstract": "Discrete diffusion has become a leading framework for generative modeling in various applications including language, vision, and biology. Existing convergence theory, however, exhibits fundamental limitations. KL-based analyses diverge under singular priors such as the masked distribution, while bounds in total variation (TV) depend on the vocabulary size $S$ and become vacuous for modern language tasks, where vocabularies contain hundreds of thousands of tokens. We develop a unified adjoint-equation-based framework that establishes vocabulary-size-independent convergence guarantees in any integral probability metric (IPM). To the best of our knowledge, our bounds are the first to be entirely free of $S$ and applicable to both masked and uniform priors. Importantly, our results can extend existing step complexity guarantees to any IPM. Also, our theory relies only on a single standard rate-matrix regularity assumption and applies to general priors. Five novel techniques drive our improvements: 1. working in the space of observables via adjoint equations rather than directly with probability measures; 2. a regularity analysis that yields bounds on any IPM; 3. a coupling argument that removes $S$-dependence under uniform transitions; and 4. score-marginal cancellation and 5. exit-routing techniques that remove $S$-dependence under masked transitions. Our framework thus sharply departs from prior analyses and avoids the shortcomings of pathspace-KL and existing TV-based approaches. Beyond convergence bounds, our framework provides a versatile toolkit for further theoretical study of discrete diffusion models, including principled choices of loss functions and vocabulary-size-independent step complexity.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.17232",
  "author_total": 6,
  "authors": [
    {
      "name": "Kelvin Kan"
    },
    {
      "name": "Xingjian Li"
    },
    {
      "name": "Benjamin J. Zhang"
    },
    {
      "name": "Tuhin Sahai"
    },
    {
      "name": "Stanley Osher"
    },
    {
      "name": "Markos A. Katsoulakis"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8282sgzw7j024.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.17232",
  "fulltext_url": null,
  "id": "8282sgzw7j024",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.17232",
  "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",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "1702|Topic Modeling"
    ]
  ],
  "openalex": "W7161724837",
  "pdf_url": "https://arxiv.org/pdf/2605.17232",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8282sgzw7j024.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:57Z",
      "native_id": "2605.17232",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.17232"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Vocabulary-size-independent Convergence of Discrete Diffusion Models: adjoint equations induce the right space",
  "title_text": "Vocabulary-size-independent Convergence of Discrete Diffusion Models: adjoint equations induce the right space",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:07Z",
  "url": "https://research.draconitic.com/paper/8282sgzw7j024",
  "venue": "arXiv",
  "versions": []
}