{
  "abstract": "We study the free-energy gradient flow associated with the depth-$N$ deep linear network on the space of invertible real $d\\times d$ matrices. The loss observes only the diagonal, \\[ E_d(W)=\\frac12\\sum_{i=1}^d(W_{ii}-1)^2, \\] and the regularizer is the Boltzmann entropy of the balanced factorization fiber computed by Menon and Yu. This is a natural higher-dimensional version of a matrix-completion problem posed by Menon as a test of whether entropy selects among a noncompact family of minimizers. For diagonal completion, every width $d\\geq2$, depth $N>2$, and inverse temperature $β>0$, we prove that the free energy is unbounded below and has exactly $2^d$ full-rank critical points, one in each diagonal sign chamber. Every critical point is diagonal and hyperbolic. Its unstable dimension is $\\binom d2$ and its stable dimension is $d(d+1)/2$. We also prove that a full-rank trajectory cannot converge to a finite rank-deficient matrix. Consequently, almost every full-rank initial condition has an unbounded forward orbit. Thus finite-temperature fiber entropy does not provide an equilibrium selection principle for diagonal completion.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.05551",
  "author_total": 1,
  "authors": [
    {
      "name": "Rodrigo Treviño"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/18j5k0sxasws0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.05551",
  "fulltext_url": null,
  "id": "18j5k0sxasws0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.05551",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DS"
    ],
    [
      "openalex",
      "1702|Stochastic Gradient Optimization Techniques"
    ]
  ],
  "openalex": "W7220664043",
  "pdf_url": "https://arxiv.org/pdf/2610.05551",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/18j5k0sxasws0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.05551",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.05551"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Failure of entropic selection for diagonal matrix completion in deep linear networks",
  "title_text": "Failure of entropic selection for diagonal matrix completion in deep linear networks",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/18j5k0sxasws0",
  "venue": "arXiv",
  "versions": []
}