{
  "abstract": "We resolve the threshold part of Question 4 of the COLT 2025 open problem \"Data Selection for Regression Tasks\" of Hanneke, Moran, Shlimovich and Yehudayoff. We study vector-valued linear regression with square loss $\\ell_{(x,y)}(W)=\\lVert Wx-y\\rVert_2^2$, where $x\\in\\mathbb{R}^d$ and $y\\in\\mathbb{R}^m$. The learner returns the minimum-Frobenius-norm empirical risk minimizer. We prove that the minimal budget of weighted examples for recovering the full-data loss on every finite dataset is exactly $n^{\\star}(d,m)=(m+1)d$. We determine the weighted selection profile $F_{\\mathrm{weighted}}(d,m,n)$ at the near-threshold budget: $F_{\\mathrm{weighted}}(d,m,(m+1)d-1)=1+1/(dm^2)$. We recover the known spanning-budget value $F_{\\mathrm{weighted}}(d,m,d)=d+1$ for every $m$, and $F_{\\mathrm{weighted}}(d,m,n)=\\infty$ for $n<d$. For the smallest open intermediate cell $(d,m)=(2,2)$ we prove $F_{\\mathrm{weighted}}(2,2,3)\\in[13/8,15/8]$ and $F_{\\mathrm{weighted}}(2,2,4)\\in[5/4,3/2]$. We reduce the conjectured exact values $13/8$ and $5/4$ to a finite moment problem on the circle with at most seven atoms and assemble structural evidence for it. The upper bounds use a fixed-basis conic compression lemma, a determinant--facet rigidity theorem for maximal certificates, and sharp sparsification lemmas for zero-mean weighted point systems. These tools may be of independent interest. We also exhibit an explicit six-point integer dataset with $d=m=2$ on which no weighted selection of $2d$ points recovers the optimal loss. Thus the scalar sufficient budget $2d$ does not extend to vector-valued outputs. Our new regression-profile results for $m\\ge2$ extend the scalar theory for $m=1$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.30254",
  "author_total": 1,
  "authors": [
    {
      "name": "Guangjian Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/48h3fhkeehzc2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.30254",
  "fulltext_url": null,
  "id": "48h3fhkeehzc2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.30254",
  "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",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "1702|Machine Learning and Algorithms"
    ]
  ],
  "openalex": "W7204868368",
  "pdf_url": "https://arxiv.org/pdf/2608.30254",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/48h3fhkeehzc2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:50Z",
      "native_id": "2608.30254",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.30254"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Exact Recovery Thresholds for Weighted Data Selection in Vector-Valued Linear Regression",
  "title_text": "Exact Recovery Thresholds for Weighted Data Selection in Vector-Valued Linear Regression",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:56Z",
  "url": "https://research.draconitic.com/paper/48h3fhkeehzc2",
  "venue": "arXiv",
  "versions": []
}