{
  "abstract": "Helmke and Shayman showed that, for generic targets, least-squares distance to the manifold of rank-r matrices is a nondegenerate Morse function whose indices are governed by Bruhat-cell codimensions. We develop the analogous theory for Monarch matrices, products of a fixed permutation matrix with two block-diagonal factors. The parametrization's gauge dimension, as well as the codimensions and closure relations of the associated Bruhat-type cells, is determined by a common transfer-matrix invariant, although these quantities are generally distinct. In the equal-block case, we prove the Monarch-Schur theorem, characterizing the permutations that minimize gauge freedom and maximize manifold dimension, and derive a general lower bound for arbitrary partitions. We further establish generic second-order nondegeneracy, a Morse-index formula, and an explicit focal set of measure zero. Finally, we show that the unregularized gradient flow lacks coercivity because of unbounded flat directions. A quartic regularizer restores global convergence, selects a canonical minimum-norm exact solution when one exists, and otherwise yields a well-defined best-approximation image. The regularizer preserves the Monarch architecture and provides a geometric interpretation of weight decay in structured factorized layers.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.02226",
  "author_total": 1,
  "authors": [
    {
      "name": "Shintaro Yoshizawa"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8bktstsd8q66h.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.02226",
  "fulltext_url": null,
  "id": "8bktstsd8q66h",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.02226",
  "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.OC"
    ],
    [
      "openalex",
      "1703|Matrix Theory and Algorithms"
    ]
  ],
  "openalex": "W7220444152",
  "pdf_url": "https://arxiv.org/pdf/2610.02226",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8bktstsd8q66h.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:50Z",
      "native_id": "2610.02226",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.02226"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "The geometry of monarch matrix approximation",
  "title_text": "The geometry of monarch matrix approximation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:50Z",
  "url": "https://research.draconitic.com/paper/8bktstsd8q66h",
  "venue": "arXiv",
  "versions": []
}