{
  "abstract": "Recursive training of generative models on their own outputs can lead to model collapse, a compounding drift away from the true data distribution. Existing theoretical works bound finite-round error accumulation in the context of diffusion models, but two questions remain open:~what distribution does the recursion converge to, and how fast? We answer both, isolating a mechanism distinct from imperfect learning: even with perfect score estimation and exact sampling, the early stopping of the reverse diffusion (required for numerical stability) drives a progressive drift away from the data distribution. We prove that this recursion converges geometrically to a unique limiting distribution, which admits a closed-form characterization as an infinite mixture of increasingly Gaussian-smoothed versions of the data distribution. A Hermite spectral decomposition of this limit reveals that recursive training acts as a low-pass filter: higher-order modes, which encode fine non-Gaussian structure, are attenuated much more strongly than coarse modes. This spectral picture motivates annealed truncation schedules that progressively shrink truncation times across retraining rounds; we prove that any schedule converging to $0$ asymptotically eliminates recursive compounding. Finally, we show our idealized characterization is robust: in the presence of discretization and score estimation errors, the learned distribution remains in a Wasserstein-2 ball around the ideal limit, with mode-dependent contraction rates that contract high-order errors faster than low-order ones. We validate the theory on synthetic Gaussian mixtures and CIFAR-10.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.13796",
  "author_total": 3,
  "authors": [
    {
      "name": "Naïl B. Khelifa"
    },
    {
      "name": "Richard E. Turner"
    },
    {
      "name": "Ramji Venkataramanan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ffq1xsxkh58sz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.13796",
  "fulltext_url": null,
  "id": "ffq1xsxkh58sz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.13796",
  "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",
      "1707|Generative Adversarial Networks and Image Synthesis"
    ]
  ],
  "openalex": "W7164866318",
  "pdf_url": "https://arxiv.org/pdf/2606.13796",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ffq1xsxkh58sz.ris",
  "sources": [
    {
      "first_seen": "2026-11-06T18:21:40Z",
      "native_id": "2606.13796",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.13796"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Recursively Trained Diffusion Models: Limiting Collapse Distribution and Spectral Characterization",
  "title_text": "Recursively Trained Diffusion Models: Limiting Collapse Distribution and Spectral Characterization",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:50Z",
  "url": "https://research.draconitic.com/paper/ffq1xsxkh58sz",
  "venue": "arXiv",
  "versions": []
}