{
  "abstract": "We study the geometry and closure of spaces of functions realized by ReLU neural networks of fixed architecture. We first establish a general result about limits of continuous piecewise-linear (CPWL) functions: the minimum number of convex sets on which a function is affine is lower semicontinuous under pointwise convergence. The proof uses a connection to hypergraph coloring and a compactness principle, and yields quantitative bounds on the polyhedral complexity of continuous limits. We then characterize the functions realizable by shallow ReLU networks with vector-valued outputs, extending a known scalar-valued characterization. Using this and the geometry of limiting hyperplane arrangements, we prove that fixed-width shallow scalar ReLU realization spaces are closed in the weighted supremum norm on the whole input space. In contrast, we construct an explicit sequence of networks with architecture $(2,3,2\\mid1)$ whose limit is not realizable by that architecture. Non-closedness holds both globally in weighted supremum norm and under uniform convergence on compact sets with nonempty interior, disproving a conjecture of Petersen, Raslan, and Voigtlaender. Finally, we show that non-closedness for a fixed scalar-output ReLU architecture necessarily involves a \\emph{blow-up residue}, in which diverging output coefficients amplify an emerging linear dependence among hidden features. Together, these results provide geometric and functional-analytic tools for studying the boundaries and degenerations of ReLU realization spaces.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.33779",
  "author_total": 1,
  "authors": [
    {
      "name": "Kathryn A. Lindsey"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/eer8gapqt3wwy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.33779",
  "fulltext_url": null,
  "id": "eer8gapqt3wwy",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.33779",
  "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",
      "cs.LG"
    ],
    [
      "arxiv",
      "math.FA"
    ],
    [
      "arxiv",
      "math.GN"
    ],
    [
      "arxiv",
      "math.MG"
    ],
    [
      "openalex",
      "1702|Neural Networks and Applications"
    ]
  ],
  "openalex": "W7223926646",
  "pdf_url": "https://arxiv.org/pdf/2610.33779",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/eer8gapqt3wwy.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.33779",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.33779"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.GN",
      "subfield": "geometry-topology"
    }
  ],
  "title": "The Geometry of Limits in ReLU Function Spaces",
  "title_text": "The Geometry of Limits in ReLU Function Spaces",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/eer8gapqt3wwy",
  "venue": "arXiv",
  "versions": []
}