{
  "abstract": "We study the width required for a randomly initialized hidden layer of a neural network to achieve rank lifting. Namely, given a dataset $X \\in \\mathbb{R}^{m \\times d}$ of $m$, $d$-dimensional input vectors separated by an angle of at least $θ$, we consider the random feature matrix $σ(XR)$, where $R$ is standard Gaussian. For positively homogeneous nonpolynomial activations, which include sign, Heaviside, ReLU, and ReLU powers among others, we prove that $$n \\gtrsim \\frac{1}θ\\max\\left\\{m,\\log\\left(\\frac{1}δ\\right)\\right\\}$$ neurons suffice for $σ(XR)$ to have full row rank $m$ with probability at least $1-δ$. This dimension-free bound exponentially improves the previous general-dimensional guarantee for sign features (Drago et al., 2026) and is essentially tight. The proof shows that one random feature column escapes every proper subspace of $\\mathbb{R}^m$ with probability $Ω(θ)$, using a coupling of nearby Gaussian directions and a local crossing of the induced hyperplane arrangement. We also study stable rank lifting, where the goal is to establish a quantitative analogue of exact rank lifting, i.e., a lower bound on the smallest eigenvalue of the empirical feature Gram matrix in high-probability. Our analysis unifies and generalizes stable rank guarantees for all $q$-homogeneous non-polynomial activations following prior work in Panigrahi et al. (2020) and Song (2026). In particular, we combine a diagonally dominant Taylor tail of the population kernel with truncation and matrix concentration, to show that for positively homogeneous nonpolynomial activations, stable rank lifting is achieved at width $$n \\gtrsim C^q \\frac{m}{θ^{2q+1}} \\log^{2q+\\frac{1}{2}}\\left(\\frac{m}θ\\right) \\log\\left(\\frac{m}δ\\right),$$ where $q$ is the degree of the activation and $C > 0$ is some universal constant.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.39855",
  "author_total": 3,
  "authors": [
    {
      "name": "Luca Becchetti"
    },
    {
      "name": "Matteo Russo"
    },
    {
      "name": "Ruben Skorupinski"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ey04e19gwywd8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.39855",
  "fulltext_url": null,
  "id": "ey04e19gwywd8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.39855",
  "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",
      "math.PR"
    ],
    [
      "openalex",
      "1702|Stochastic Gradient Optimization Techniques"
    ]
  ],
  "openalex": "W7215081743",
  "pdf_url": "https://arxiv.org/pdf/2609.39855",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ey04e19gwywd8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2609.39855",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.39855"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Dimension-Free Rank Lifting from Random Hyperplane Arrangements",
  "title_text": "Dimension-Free Rank Lifting from Random Hyperplane Arrangements",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/ey04e19gwywd8",
  "venue": "arXiv",
  "versions": []
}