{
  "abstract": "For $d \\geq 2$, $p \\geq 1$ and $ε> 0$, let $N_p(d,ε)$ be the smallest integer $N$ such that for every integer $n$ and every $A\\in\\mathbb{R}^{n\\times d}$, there exists a matrix $Φ\\in\\mathbb{R}^{N\\times n}$ satisfying $(1-ε)\\lVert Ax\\rVert_p\\leq \\lVertΦA x\\rVert_p\\leq (1+ε)\\lVert Ax\\rVert_p$ for all $x\\in\\mathbb{R}^d$. For every constant $p\\geq 1$ with $p\\not\\in 2\\mathbb{Z}$, when $d\\gtrsim_p \\log(1/ε)$, the bound \\[ N_p(d,ε) \\gtrsim_{p} \\frac{d}{ε^2 \\operatorname{polylog}(d/ε)} \\] is established. This improves the previous lower bound $Ω(1/(ε^2\\operatorname{polylog}(1/ε)))$ due to Li et al. (SICOMP 2021) and is optimal up to logarithmic factors for $1\\leq p<2$. The central technical idea originated from ChatGPT 6 Pro.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.14201",
  "author_total": 1,
  "authors": [
    {
      "name": "Yi Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1t5bf53ekb9pa.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.14201",
  "fulltext_url": null,
  "id": "1t5bf53ekb9pa",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.14201",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.DS"
    ],
    [
      "arxiv",
      "math.FA"
    ],
    [
      "openalex",
      "2206|Sparse and Compressive Sensing Techniques"
    ]
  ],
  "openalex": "W7203603527",
  "pdf_url": "https://arxiv.org/pdf/2608.14201",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1t5bf53ekb9pa.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2608.14201",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.14201"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "A Near-Optimal Lower Bound for $\\ell_p$-Subspace Embeddings, $1\\leq p<2$",
  "title_text": "A Near-Optimal Lower Bound for ℓ_p-Subspace Embeddings, 1≤ p<2",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/1t5bf53ekb9pa",
  "venue": "arXiv",
  "versions": []
}