{
  "abstract": "For two Gaussian random elements on a separable Hilbert space with common covariance operator, the fourth-order moment tensor structure of their mixture admits an operator-valued representation obtained from the purely quadratic component of the class-conditional second moment. We identify the tensor mechanism generating the completely diagonal coefficients of this fourth-order representation after covariance standardization, and show that these coefficients are governed entirely by the coordinatewise Cameron-Martin energy of the mean shift. This yields a necessary and sufficient condition, together with an explicit eigenvalue, for the Fisher discriminant to be an eigenfunction of the induced coordinatewise operator. We further introduce aggregated fourth-order spectral functionals for two complementary constructions, one built from fourth-order moments and the other from a product-of-expectations counterpart, and prove that they are asymptotically equivalent precisely when the underlying Gaussian measures are mutually singular. These results provide a spectral realization of the classical Cameron-Martin criterion through higher-order moment operators, explaining the probabilistic origin of fourth-order spectral quantities previously proposed for Gaussian discrimination and functional data classification and relating them to the \"near-perfect classification\" regime.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.30733",
  "author_total": 1,
  "authors": [
    {
      "name": "Marc Vidal"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1jsc5tk16f9ds.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.30733",
  "fulltext_url": null,
  "id": "1jsc5tk16f9ds",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.30733",
  "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.PR"
    ],
    [
      "arxiv",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "2613|Advanced Statistical Methods and Models"
    ]
  ],
  "openalex": "W7204972060",
  "pdf_url": "https://arxiv.org/pdf/2608.30733",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1jsc5tk16f9ds.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2608.30733",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.30733"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "An operator-theoretic characterization of Gaussian measure singularity under mean shifts",
  "title_text": "An operator-theoretic characterization of Gaussian measure singularity under mean shifts",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:33Z",
  "url": "https://research.draconitic.com/paper/1jsc5tk16f9ds",
  "venue": "arXiv",
  "versions": []
}