{
  "abstract": "We develop a general framework for analyzing representation costs induced by parameter-space regularizers in data-fitting methods. For an arbitrary parametric method, we define its representation cost and native function space, prove existence, and identify conditions under which parameter-space and function-space problems have equal infimal values and minimizers transfer between them. This framework yields representer theorems and recovers classical formulations---including kernel methods and RKHSs, wavelets and Besov spaces, and shallow neural networks and variation spaces---as special cases. Our main new results concern depth-$L$ feedforward ReLU networks with weight-decay regularization. For these networks, we prove that the representation cost is a power of a quasi-seminorm and that, under suitable hypotheses, the native space is a quasi-Banach space with nonconvex unit ball when $L > 2$. These results identify a novel depth-dependent quasi-Banach geometry induced by weight decay.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.14954",
  "author_total": 2,
  "authors": [
    {
      "name": "Greg Ongie"
    },
    {
      "name": "Rahul Parhi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1dpy2gv5sc5xr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.14954",
  "fulltext_url": null,
  "id": "1dpy2gv5sc5xr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.14954",
  "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.OC"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "openalex",
      "1702|Stochastic Gradient Optimization Techniques"
    ]
  ],
  "openalex": "W7164884728",
  "pdf_url": "https://arxiv.org/pdf/2606.14954",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1dpy2gv5sc5xr.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:57Z",
      "native_id": "2606.14954",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.14954"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Representation Costs in Data Science: Foundations and the Quasi-Banach Spaces of Deep Neural Networks",
  "title_text": "Representation Costs in Data Science: Foundations and the Quasi-Banach Spaces of Deep Neural Networks",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:07Z",
  "url": "https://research.draconitic.com/paper/1dpy2gv5sc5xr",
  "venue": "arXiv",
  "versions": []
}