{
  "abstract": "Bubeck and Sellke's universal law of robustness says that fitting $n$ noisy samples below the noise level with a class of $p$ real parameters forces a Lipschitz constant of order $\\sqrt{nd/p}$, when the covariate distribution on $\\mathbb{R}^d$ is isoperimetric: smooth interpolation costs a factor $d$ of overparametrization. Its hypotheses are Gaussian-type concentration, a homogeneous covariate law, scalar outputs, and independent samples; we relax each one. Exponential concentration suffices, at the unchanged rate and absolute constant and at the price of a capacity burn-in, some form of which we show is necessary; this gives the law for normalized log-concave covariates, with a $\\log d$ loss from the current KLS bound. For heterogeneous mixtures the worst component constant is replaced by the weighted average of the component constants, under a typicality condition on the mixture composition. For vector-valued predictors the bound at the Frobenius Lipschitz scale is free of the output dimension $K$, and the Euclidean bound loses only $K^{-1/2}$, a loss that covariate-independent noise removes. Stationary Markov covariates retain the i.i.d. rate, the pseudo-spectral gap entering only the sample-size conditions. One master theorem carries the shared argument, and constants are explicit throughout.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.03653",
  "author_total": 1,
  "authors": [
    {
      "name": "Tobias J. Peter"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2dwksrx11wcgd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.03653",
  "fulltext_url": null,
  "id": "2dwksrx11wcgd",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.03653",
  "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.PR"
    ],
    [
      "arxiv",
      "stat.ML"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.03653",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2dwksrx11wcgd.ris",
  "sources": [
    {
      "first_seen": "2026-11-04T15:21:29Z",
      "native_id": "2611.03653",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.03653"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Four Extensions of the Universal Law of Robustness: Log-Concave Covariates, Heterogeneous Mixtures, Vector-Valued Predictors, and Markovian Data",
  "title_text": "Four Extensions of the Universal Law of Robustness: Log-Concave Covariates, Heterogeneous Mixtures, Vector-Valued Predictors, and Markovian Data",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:10Z",
  "url": "https://research.draconitic.com/paper/2dwksrx11wcgd",
  "venue": "arXiv",
  "versions": []
}