{
  "abstract": "Training data constrains optimizer geometry through the covectors visible to a declared information channel. We study how such partial information determines a full positive cometric relative to a reference and which degrees of freedom remain unidentified. Our central result resolves full-column-rank positive-definite compression under affine-invariant Riemannian geometry. The compression map is a split-Hadamard metric submetry and admits an explicit unique completion that is the affine-invariant nearest full geometry realizing a visible target and yields exact full-to-visible variational reduction. When the channel moves, the completions form a gauge-invariant rank stratification of the positive-definite cone. Its closed-form pullback pair metric separates visible-metric motion from subspace rotation through a reference-mismatch weight, yields an explicit positive-semidefinite multi-direction Gram matrix, and exposes the precise singularity of reference-valued modes. The mechanism is explained by a metric theorem equating ball submetry, attained fiber distance, and lossless reduction of every monotone radial visible decision problem. A smooth split-Hadamard theorem supplies coherent information sheets, proximal commutation, and solution-wise gradient-flow lifting. The positive-definite realization also gives closed-form prior-data shrinkage. Diagonal and block optimizer families reduce to relative-interior conic image tests with valid facial certificates, while deterministic and finite-sample bounds quantify recovery of the visible geometry and its subspace. Together these results characterize exact reduction, reference-dependent completion, and structured expressivity for the stated finite-dimensional affine-invariant model.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.12991",
  "author_total": 1,
  "authors": [
    {
      "name": "Zavier Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/31mr2yna4f6ht.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.12991",
  "fulltext_url": null,
  "id": "31mr2yna4f6ht",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.12991",
  "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.LG"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "1702|Stochastic Gradient Optimization Techniques"
    ]
  ],
  "openalex": "W7212725155",
  "pdf_url": "https://arxiv.org/pdf/2609.12991",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/31mr2yna4f6ht.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:11Z",
      "native_id": "2609.12991",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.12991"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Information-Induced Training Geometry: Exact Reduction, Canonical Completion, and Structured Expressivity",
  "title_text": "Information-Induced Training Geometry: Exact Reduction, Canonical Completion, and Structured Expressivity",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:11Z",
  "url": "https://research.draconitic.com/paper/31mr2yna4f6ht",
  "venue": "arXiv",
  "versions": []
}