{
  "abstract": "We prove that fully independent SparseStack achieves the oblivious subspace embedding parameters conjectured by Nelson and Nguyen (FOCS 2013): $m=O((d+\\log(1/δ))/\\varepsilon^2)$ rows and $s=O(\\log(d/δ)/\\varepsilon)$ nonzero entries per column for distortion $\\varepsilon$ and failure probability $δ$ on any fixed $d$-dimensional subspace, with explicit constants. The proof turns random-matrix concentration into a problem in finite-dimensional linear algebra. A coupling first reduces the moment estimates to a model with independent finite-valued entries. We represent these variables by multiplication operators, so their matrix moments become exact matrix elements of a deterministic operator on a finite tensor product. The central estimate bounds the contribution of $\\ell\\ge1$ occupied sites sharing the external factor $\\mathbb{R}^d$ by $d+\\ell-1$ rather than $d\\ell$, yielding additive dependence on the dimension and the moment order. This approach controls both spectral edges without Gaussian comparison. The main theorem has been formally verified in Lean 4.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.02978",
  "author_total": 1,
  "authors": [
    {
      "name": "Diar Heidary"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dq5tx66vxkq7y.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.02978",
  "fulltext_url": null,
  "id": "dq5tx66vxkq7y",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.02978",
  "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.DS"
    ],
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "openalex",
      "2206|Sparse and Compressive Sensing Techniques"
    ]
  ],
  "openalex": "W7208793922",
  "pdf_url": "https://arxiv.org/pdf/2609.02978",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dq5tx66vxkq7y.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:15Z",
      "native_id": "2609.02978",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.02978"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "SparseStack Is an Optimal Oblivious Subspace Embedding",
  "title_text": "SparseStack Is an Optimal Oblivious Subspace Embedding",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:15Z",
  "url": "https://research.draconitic.com/paper/dq5tx66vxkq7y",
  "venue": "arXiv",
  "versions": []
}