{
  "abstract": "Deep representation learning often selects hidden features and fits the final predictor on the same sample, so fixed-feature analysis performed after selection can omit selection cost. We study the conditional empirical Rademacher complexity of deep ReLU representations followed by bounded-norm predictors in additive or Lévy-Brownian RKHSs, termed Brownian heads. For a fixed representation, we derive an exact dual identity and sharp bounds in terms of activation mass, the average norm of the observed hidden vectors. Under same-sample selection, the representation supremum induces a quadratic Rademacher process. Brownian layer-cake and Gaussian-projection identities reduce it to coordinatewise or signed projected threshold traces, separating realized scale from selection complexity. For samples with pairwise-distinct inputs, explicit scalar ReLU families match the finite-trace and VC rates up to universal constants at the realized trace-and-envelope level. Induced-norm contraction also yields architecture-level bounds for rectangular, rank-deficient ReLU networks. Experiments verify the sharp bounds and rates, exhibit a selection gap at fixed activation mass, and assess the predictive feasibility of Brownian heads.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.21422",
  "author_total": 2,
  "authors": [
    {
      "name": "Mahdi Mohammadigohari"
    },
    {
      "name": "Nicole Mücke"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/chm61ybg87kra.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.21422",
  "fulltext_url": null,
  "id": "chm61ybg87kra",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.21422",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "openalex",
      "1702|Stochastic Gradient Optimization Techniques"
    ]
  ],
  "openalex": "W7213931859",
  "pdf_url": "https://arxiv.org/pdf/2609.21422",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/chm61ybg87kra.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:12Z",
      "native_id": "2609.21422",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.21422"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Brownian Heads for Deep ReLU Representations: Activation Mass and the Cost of Same-Sample Selection",
  "title_text": "Brownian Heads for Deep ReLU Representations: Activation Mass and the Cost of Same-Sample Selection",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:12Z",
  "url": "https://research.draconitic.com/paper/chm61ybg87kra",
  "venue": "arXiv",
  "versions": []
}