{
  "abstract": "We prove an explicit lower spectral bound for independent random vectors with prescribed group multiplicities. Let $v_{ij}\\in\\mathbb C^d$, $1\\le i\\le l$, $1\\le j\\le k_i$, be independent random vectors with finite support and $\\mathbb E v_{ij}v_{ij}^*=A_i$, where $\\sum_{i=1}^l A_i=I_d$ and $\\operatorname{Tr}(A_i)\\geε$. Set $k=\\max_i k_i$, $n=\\max_i\\operatorname{rank}(A_i)$, and $K=\\sum_{i=1}^l k_i$. If $n>(6+4\\sqrt2)k$ and $2\\sqrt{nk}-k<ε\\le n/2$, then, with positive probability, \\[ λ_K\\left(\\sum_{i=1}^l\\sum_{j=1}^{k_i}v_{ij}v_{ij}^*\\right) \\ge n-(\\sqrt{k}+\\sqrt{n-ε})^2. \\] As a consequence, we obtain a restricted invertibility result selecting exactly $k_i$ vectors from the $i$th group of a Parseval frame. The proof combines interlacing families, a reciprocal transformation of mixed characteristic polynomials, and a multivariate barrier estimate that accommodates repeated differentiation in the same variable.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.15583",
  "author_total": 2,
  "authors": [
    {
      "name": "Xinyu Chen"
    },
    {
      "name": "Bochen Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6h2g3bf9n6na7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.15583",
  "fulltext_url": null,
  "id": "6h2g3bf9n6na7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.15583",
  "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",
      "math.FA"
    ],
    [
      "openalex",
      "2604|Mathematical Analysis and Transform Methods"
    ]
  ],
  "openalex": "W7221768178",
  "pdf_url": "https://arxiv.org/pdf/2610.15583",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6h2g3bf9n6na7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.15583",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.15583"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Explicit Bounds for Restricted Invertibility with Prescribed Group Cardinalities",
  "title_text": "Explicit Bounds for Restricted Invertibility with Prescribed Group Cardinalities",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/6h2g3bf9n6na7",
  "venue": "arXiv",
  "versions": []
}