{
  "abstract": "We study the minimax estimation error for distributed covariance matrix estimation in the vertical-split (feature-split) setting, where two agents each observe different coordinates of~$m$ i.i.d.\\ sub-Gaussian samples and communicate a limited number of bits to a central server. While \\cite{rahmani2025fundamental} established nearly tight bounds for dense (unstructured) cross-covariance matrices, we investigate whether imposing elementwise $s$-sparsity on the cross-covariance $C_{21}$ can reduce the required communication and sample complexity. We show that sparsity can reduce the communication required for cross-covariance estimation in the vertical split at a fixed target distortion. Specifically, for sufficiently large $d_1d_2/s'$ and $0<\\varepsilon<σ^2\\sqrt{s'}/32$, any scheme achieving expected Frobenius distortion at most $\\varepsilon$ must satisfy $B_k = Ω(σ^4 d_k\\, s' \\log(d_1 d_2/s')/\\varepsilon^2)$ and $m = Ω(σ^4\\, s' \\log(d_1 d_2/s')/\\varepsilon^2)$ for cross-covariance estimation, where $s' = s \\wedge d_{\\min}$. For the $1$-sparse case, our achievable scheme reduces the $d_1d_2$ factor in the dense communication rate to $\\log(d_1d_2)$, up to polylogarithmic factors, for the cross-covariance communication component in the matching regime. Our lower bounds are established via Fano's method with an explicit sparse packing using a Varshamov--Gilbert-type argument for signed partial permutation matrices combined with the Conditional Strong Data Processing Inequality of \\cite{rahmani2025fundamental}. We show that the communication lower bound is tight up to polylogarithmic factors under the conditions of Remark 4.9, using an achievable scheme based on covering-net quantization and entry-wise hard thresholding.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.07124",
  "author_total": 2,
  "authors": [
    {
      "name": "Jing Yee Tan"
    },
    {
      "name": "Guangyue Han"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/58vn0pe64fpmj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.07124",
  "fulltext_url": null,
  "id": "58vn0pe64fpmj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.07124",
  "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.IT"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "openalex",
      "2613|Statistical Methods and Inference"
    ]
  ],
  "openalex": "W7163953942",
  "pdf_url": "https://arxiv.org/pdf/2606.07124",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/58vn0pe64fpmj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:57Z",
      "native_id": "2606.07124",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.07124"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Information-Theoretic Bounds for Sparse Covariance Estimation in the Vertical-Split Distributed Model",
  "title_text": "Information-Theoretic Bounds for Sparse Covariance Estimation in the Vertical-Split Distributed Model",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:03Z",
  "url": "https://research.draconitic.com/paper/58vn0pe64fpmj",
  "venue": "arXiv",
  "versions": []
}