{
  "abstract": "We address the Subset selection problem for matrices, where the goal is to select a subset $\\mathcal{S}$ of $k$ column indices from a \\enquote{short-and-fat} matrix $X \\in \\mathbb{R}^{m \\times n}$, such that the sampled submatrix $X_{\\mathcal{S}}$ has $\\|X_{\\mathcal{S}}^†X\\|_F$ as small as possible. Our approach is centered on volume sampling, which attains the tightest known bound on this objective in expectation. As our primary contribution, we propose a new deterministic algorithm, Forward derandomized volume sampling (FDVS), which provably attains this bound and has asymptotic complexity $O(nkm)$. In contrast, the complexity of all previously known algorithms with this guarantee scales at least quadratically in $n$ when $k \\ll n$, making FDVS particularly attractive for very wide matrices. In addition, we systematically structure the landscape of volume sampling methods: we propose a simple $O(nm^2)$ algorithm for exact forward volume sampling, show that a known deterministic method is in fact a derandomization of Reverse iterative volume sampling, and derive a modification of the latter that avoids repeated SVD downdating, along with a fast greedy forward variant. The algorithms are verified and compared in numerical experiments involving optimal experimental design, sensor placement, and DLRA-DEIM.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.08576",
  "author_total": 2,
  "authors": [
    {
      "name": "Ivan Kozyrev"
    },
    {
      "name": "Alexander Osinsky"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0arvprdca0kzd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.08576",
  "fulltext_url": null,
  "id": "0arvprdca0kzd",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.08576",
  "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.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "openalex",
      "1703|Matrix Theory and Algorithms"
    ]
  ],
  "openalex": "W7220962367",
  "pdf_url": "https://arxiv.org/pdf/2610.08576",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0arvprdca0kzd.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:26Z",
      "native_id": "2610.08576",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.08576"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Subset selection for matrices by volume sampling",
  "title_text": "Subset selection for matrices by volume sampling",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:26Z",
  "url": "https://research.draconitic.com/paper/0arvprdca0kzd",
  "venue": "arXiv",
  "versions": []
}