{
  "abstract": "Many parameter-efficient methods generate the parameters of a large neural network from a low-dimensional latent representation. Given an architecture $Φ$ with $P_Φ$ parameter slots, we write $\\boldsymbolθ_f=\\mathcal{G}(\\boldsymbolξ_f)$, where $\\mathcal{G}\\colon\\mathbb{R}^M\\to\\mathbb{R}^{P_Φ}$ is a parameter generator and $\\boldsymbolξ_f\\in\\mathbb{R}^M$ is a latent representation of the target function $f$. The architecture $Φ$ and the generator $\\mathcal{G}$ are shared across the entire target class, while each target $f$ is represented by its own latent vector $\\boldsymbolξ_f$, with $Φ_{\\mathcal{G}(\\boldsymbolξ_f)}$ approximating $f$. This framework encompasses hypernetworks, low-dimensional parameterizations, parameter-efficient adaptation, and model compression. Understanding the tradeoff between the latent dimension $M$ and the network budget $P$ is therefore fundamental to characterizing the expressive efficiency of these methods. We study this tradeoff for affine generators and fully connected ReLU architectures. More precisely, optimizing jointly over architectures $Φ$ satisfying $P_Φ\\leq P$ and affine generators $\\mathcal{G}:\\mathbb{R}^M\\to \\mathbb{R}^{P_Φ}$, we prove that the optimal worst-case uniform approximation error over the unit ball of $α$-Hölder functions on $[0,1]^d$, where $0<α\\leq1$, has the sharp order $ \\bigl(P\\min\\{M,P\\}\\bigr)^{-α/d}. $ In particular, our result shows that even a fixed-dimensional latent space suffices to achieve vanishing approximation error as the network budget increases.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.31157",
  "author_total": 1,
  "authors": [
    {
      "name": "Shijun Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0apm1q19dtjyr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.31157",
  "fulltext_url": null,
  "id": "0apm1q19dtjyr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.31157",
  "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",
      "stat.ML"
    ],
    [
      "openalex",
      "1702|Neural Networks and Applications"
    ]
  ],
  "openalex": "W7204954704",
  "pdf_url": "https://arxiv.org/pdf/2608.31157",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-31",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0apm1q19dtjyr.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:54Z",
      "native_id": "2608.31157",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.31157"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Sharp Approximation Rates for Neural Networks with Affine Latent Parameterizations",
  "title_text": "Sharp Approximation Rates for Neural Networks with Affine Latent Parameterizations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:54Z",
  "url": "https://research.draconitic.com/paper/0apm1q19dtjyr",
  "venue": "arXiv",
  "versions": []
}