{
  "abstract": "We study the projected power method (PPM) for synchronizing \\(n\\) unknown permutations of \\(m\\) objects under a possibly sparse uniform corruption model. Each pair is observed with probability \\(p\\), and an observed measurement is uncorrupted with probability \\(π_0\\) and is otherwise an independent uniform permutation. Under \\(\\log m=o(npπ_0^2)\\), we prove exact one-step recovery (with high probability) of each prescribed block for an independent estimate with a fixed positive majority of correct blocks. When \\(np\\ge C_0\\log n\\) and \\(m=o(npπ_0^2)\\), we prove that one high-probability event yields a block-error contraction simultaneously for every estimate whose optimally aligned error is at most \\(0.5-ε\\). The contraction factor is \\(O(m/(npπ_0^2))\\) and the error floor is \\(O(e^{-cnpπ_0}+e^{-cnpπ_0^2}+{\\log n}/{n})\\). Consequently, one update maps every possibly data-dependent estimate in this basin to vanishing block error, and all subsequent iterates remain almost exact uniformly over the iteration index. The one-step and trajectory results extend to independent, non-identically distributed, permutation-valued corruptions with mean \\(m^{-1} \\mathbf{1}\\mathbf{1}^{\\top}\\). Under the uniform model, a reference-block spectral initializer has aligned block error \\(O_{\\mathbb P}(m/(npπ_0^2))\\), yielding an end-to-end almost-exact recovery guarantee. Under a stronger all-block signal condition, PPM reaches exact recovery after finitely many iterations. The theory transfers exactly to partial permutations with common support; for varying supports, we establish deterministic and probabilistic co-visibility margins.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.09502",
  "author_total": 2,
  "authors": [
    {
      "name": "Vahan Huroyan"
    },
    {
      "name": "Gilad Lerman"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9amhttfjc04yr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.09502",
  "fulltext_url": null,
  "id": "9amhttfjc04yr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.09502",
  "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.IT"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "openalex",
      "2206|Sparse and Compressive Sensing Techniques"
    ]
  ],
  "openalex": "W7212111336",
  "pdf_url": "https://arxiv.org/pdf/2609.09502",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9amhttfjc04yr.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:15Z",
      "native_id": "2609.09502",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.09502"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Recovery Theory for Projected Power Iterations in Permutation Synchronization",
  "title_text": "Recovery Theory for Projected Power Iterations in Permutation Synchronization",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:22Z",
  "url": "https://research.draconitic.com/paper/9amhttfjc04yr",
  "venue": "arXiv",
  "versions": []
}