{
  "abstract": "Quaternion tensor singular value decomposition (T-SVD) under the T-product provides a framework for analyzing multidimensional quaternion data. However, the T-SVD construction of Pan and Ng (SIAM J. Matrix Anal. Appl., 2024) \\cite{PanNg2024} contains a structural gap: an unrestricted quaternion matrix SVD need not preserve the paired Fourier block structure required for tensor factor recovery, potentially leading to loss of T-unitarity and inaccurate reconstruction. To address this issue, we develop a structure-preserving construction that transforms each paired quaternion block into a complex matrix and maps its complex SVD factors back to quaternion factors with the required structure. This construction corrects the proof of \\cite[Theorem 4]{PanNg2024} and leads to a corrected version of \\cite[Algorithm 2]{PanNg2024}, hereafter referred to as the ``corrected algorithm'', ensuring T-unitary tensor factors and an f-diagonal core for arbitrary third-order quaternion tensors. A numerical counterexample demonstrates the failure of \\cite[Algorithm 2]{PanNg2024} and verifies the corrected algorithm at machine precision. Notably, we propose a fast structure-preserving algorithm, Algorithm~\\ref{alg:t-fast}, and compare it numerically with the corrected algorithm. The experimental results show that Algorithm~\\ref{alg:t-fast} reduces the decomposition time while maintaining near-machine-precision accuracy. Furthermore, its application to color video encryption yields high reconstruction quality, favorable ciphertext statistics, and stability under input perturbations.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.31451",
  "author_total": 3,
  "authors": [
    {
      "name": "Daochang Zhang"
    },
    {
      "name": "Yue Zhao"
    },
    {
      "name": "Jingqian Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c3mch2hrbha45.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.31451",
  "fulltext_url": null,
  "id": "c3mch2hrbha45",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.31451",
  "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.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "openalex",
      "2605|Tensor decomposition and applications"
    ]
  ],
  "openalex": "W7223338246",
  "pdf_url": "https://arxiv.org/pdf/2610.31451",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c3mch2hrbha45.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:38Z",
      "native_id": "2610.31451",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.31451"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "A Fast Algorithm for Quaternion Tensor Singular Value Decomposition under the T-product",
  "title_text": "A Fast Algorithm for Quaternion Tensor Singular Value Decomposition under the T-product",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:38Z",
  "url": "https://research.draconitic.com/paper/c3mch2hrbha45",
  "venue": "arXiv",
  "versions": []
}