{
  "abstract": "We prove that for every isotropic log-concave distribution $P$ on $\\mathbb{R}^d$ and every even $m\\ge2$, the polynomial $(Cm)^m\\|v\\|_2^m - \\mathbb{E}_{X\\sim P}\\langle X,v\\rangle^m$ is a sum of squares, where $C>0$ is a universal constant. This improves on the Poincaré-dependent bounds (Kothari and Steinhardt, 2017), recovering the optimal moment bounds for log-concave distributions. As an immediate corollary, we obtain computationally efficient algorithms with dimension-free error guarantees for a wide range of statistical estimation problems. Our proof proceeds by using stochastic localization to decompose $P$ as an average of strongly log-concave measures, whose centered moments admit the subgaussian certificates (Diakonikolas et al., 2025). With a covariance-adapted choice of localization, we show that a fourth-moment certificate derived from the variance inequality for quadratic forms (Letwin, 2026) suffices to control this averaging at every even degree.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.30105",
  "author_total": 1,
  "authors": [
    {
      "name": "Aleksandr Storozhenko"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b3sb8cbmy00nf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.30105",
  "fulltext_url": null,
  "id": "b3sb8cbmy00nf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.30105",
  "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.CC"
    ],
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2604|Point processes and geometric inequalities"
    ]
  ],
  "openalex": "W7214286925",
  "pdf_url": "https://arxiv.org/pdf/2609.30105",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b3sb8cbmy00nf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2609.30105",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.30105"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "On the SoS Certifiability of Log-Concave Distributions",
  "title_text": "On the SoS Certifiability of Log-Concave Distributions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/b3sb8cbmy00nf",
  "venue": "arXiv",
  "versions": []
}