{
  "abstract": "How much risk does a small reweighted training support retain? For finite weighted least squares with the minimum-norm learner, we prove the exact law $Γ_d(n)=3-n/d$ throughout $\\lceil3d/2\\rceil\\leq n\\leq2d-1$. The guarantee covers every observed feature rank and uses selections that preserve the full feature span. Balanced simplex anchors reduce dimension; positive-weight lifting and independent-line compression close the risk bound. Shifted coordinate pairs attain the matching lower bound. The complete dataset-level upper bound and sharpness construction are verified in Lean 4. At the smaller budget $(d,n)=(5,6)$, we also prove $Γ_5(6)=11/5$, matching the simplex-block prediction from $5=3+2$ over arbitrary interacting configurations. Circuit covers, comparison second moments, and circuit-plane probabilities give the sharp excess $6/5$, while polar-face geometry resolves shared rank-three circuits. The general simplex-block frontier connects these laws within the intermediate-budget selection problem.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.00101",
  "author_total": 2,
  "authors": [
    {
      "name": "Zhongxuan Liu"
    },
    {
      "name": "Hongzhi Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ehhh889dtjfze.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.00101",
  "fulltext_url": null,
  "id": "ehhh889dtjfze",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.00101",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "openalex",
      "1702|Machine Learning and Algorithms"
    ]
  ],
  "openalex": "W7216584435",
  "pdf_url": "https://arxiv.org/pdf/2610.00101",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ehhh889dtjfze.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:14Z",
      "native_id": "2610.00101",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.00101"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Weighted Data Selection: Sharp Upper-Half and Five-Dimensional Laws",
  "title_text": "Weighted Data Selection: Sharp Upper-Half and Five-Dimensional Laws",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:14Z",
  "url": "https://research.draconitic.com/paper/ehhh889dtjfze",
  "venue": "arXiv",
  "versions": []
}