{
  "abstract": "Large-scale tensors arising in science and engineering are often too large to form or store explicitly. Their multilinear contractions, however, can frequently be evaluated efficiently, which makes randomized Frobenius-norm estimation a practical alternative to exact computation. In this paper, we study estimation of the squared Frobenius norm of a real order-$d$ tensor from such contractions using independent real, complex, or quaternion Gaussian probes. For a rank-one tensor, one normalized sample is a product of $d-1$ independent Gamma variables. This gives exact relative-variance factors $3^{d-1}-1$, $2^{d-1}-1$, and $(3/2)^{d-1}-1$, respectively. Under the same budget of underlying real Gaussian vectors, the quaternion probe has smaller rank-one variance than four independent real probes for $d\\ge3$; a real-channel representation identifies the gain as modewise pooling. For orthogonal CP tensors, the exact relative variance equals the corresponding rank-one factor divided by the participation-ratio effective rank and the number of samples. For arbitrary real tensors, we prove sharp uniform moment bounds, attained at rank one, and obtain Chebyshev guarantees. A classical block-median construction then gives two-sided exponential confidence for the modified estimator. Numerical examples test the rank-one and effective-rank formulas. An implicit infinite-series kernel tensor illustrates contraction-based norm evaluation; EEG maps provide a qualitative spatial interpolation illustration.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.12534",
  "author_total": 3,
  "authors": [
    {
      "name": "Yannan Chen"
    },
    {
      "name": "Hailin Sun"
    },
    {
      "name": "Min Xi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/89khye1hgzw6z.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.12534",
  "fulltext_url": null,
  "id": "89khye1hgzw6z",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.12534",
  "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",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "2605|Tensor decomposition and applications"
    ]
  ],
  "openalex": "W7221545458",
  "pdf_url": "https://arxiv.org/pdf/2610.12534",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/89khye1hgzw6z.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:28:40Z",
      "native_id": "2610.12534",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.12534"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Quaternion Gaussian Probes for Tensor Frobenius-Norm Estimation",
  "title_text": "Quaternion Gaussian Probes for Tensor Frobenius-Norm Estimation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:28:46Z",
  "url": "https://research.draconitic.com/paper/89khye1hgzw6z",
  "venue": "arXiv",
  "versions": []
}