{
  "abstract": "Jiang, Lee, and Vempala conjectured that if $X,Y\\in\\mathbb{R}^n$ are independent isotropic log-concave random vectors, then $W_2(L(\\langle X,Y\\rangle),N(0,n))$ is bounded by a universal constant. Subject to Theorems 1.2 and 2.5 of arXiv:2607.24164v1, we prove this conjecture and a rectangular bilinear-form extension. For independent isotropic log-concave $X\\in\\mathbb{R}^m$, $Y\\in\\mathbb{R}^n$, and nonzero $B\\in\\mathbb{R}^{m\\times n}$, put \\[ r_4(B)=\\frac{(\\operatorname{Tr}(B^\\top B))^2} {\\operatorname{Tr}((B^\\top B)^2)}. \\] We construct a nonnegative scalar Stein kernel for $X^\\top B Y/\\|B\\|_F$ whose squared $L^2$ discrepancy is at most $20/r_4(B)$, and consequently obtain the same bound for squared $2$-Wasserstein distance to $N(0,1)$. The proof develops an exact covariance identity and deficit decomposition for trace observables of moment-map Stein kernels, together with a stability theorem for positive Stein kernels under log-concave approximation. Taking $B=I_n$ yields \\[ W_2^2\\left(L\\left(\\frac{\\langle X,Y\\rangle}{\\sqrt{n}}\\right),N(0,1)\\right)\\leq\\frac{20}{n}, \\] which is the Jiang--Lee--Vempala conjecture.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.27657",
  "author_total": 1,
  "authors": [
    {
      "name": "Tianle Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0b4we976fxcmk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.27657",
  "fulltext_url": null,
  "id": "0b4we976fxcmk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.27657",
  "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",
      "math.FA"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2613|Random Matrices and Applications"
    ]
  ],
  "openalex": "W7204780338",
  "pdf_url": "https://arxiv.org/pdf/2608.27657",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0b4we976fxcmk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:04Z",
      "native_id": "2608.27657",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.27657"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Stein Kernels and Normal Approximation for Log-Concave Bilinear Forms",
  "title_text": "Stein Kernels and Normal Approximation for Log-Concave Bilinear Forms",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:04Z",
  "url": "https://research.draconitic.com/paper/0b4we976fxcmk",
  "venue": "arXiv",
  "versions": []
}