{
  "abstract": "The phase retrieval problem involves reconstructing a function or signal solely from the magnitude of linear measurements. Most theoretical analyses of phase retrieval algorithms rely on i.i.d. Gaussian random measurements or sub-Gaussian random measurements. In this paper, our focus is on the fusion frame phase retrieval problem, where the sampling matrices are i.i.d. rank-$r$ orthogonal projections drawn from the Haar measure. We present concentration inequalities for functions on the set of rank-$r$ orthogonal projection matrices. These inequalities are crucial for the theoretical analysis of the fusion frame phase retrieval problem. Based on these inequalities, we demonstrate that gradient descent, combined with a two-stage initialization, achieves linear convergence to the target signal up to a global phase with a measurement complexity of $O(d\\log^2 d)$ when the rank $r = O(1)$. We verify this convergence through numerical results.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.08531",
  "author_total": 3,
  "authors": [
    {
      "name": "Haixia Liu"
    },
    {
      "name": "Bing Gao"
    },
    {
      "name": "Yang Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9syz8vhe118nr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.08531",
  "fulltext_url": null,
  "id": "9syz8vhe118nr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.08531",
  "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"
    ],
    [
      "openalex",
      "2206|Sparse and Compressive Sensing Techniques"
    ]
  ],
  "openalex": "W7212056674",
  "pdf_url": "https://arxiv.org/pdf/2609.08531",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9syz8vhe118nr.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.08531",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.08531"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.IT",
      "subfield": "general"
    }
  ],
  "title": "The Fusion Frame Phase Retrieval",
  "title_text": "The Fusion Frame Phase Retrieval",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/9syz8vhe118nr",
  "venue": "arXiv",
  "versions": []
}