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Dimension-Free Rank Lifting from Random Hyperplane Arrangements — arXiv:2609.39855

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arXiv (en)

Original

https://arxiv.org/abs/2609.39855

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CC0-1.0

Archived

2026-10-07T13:46:45.003Z

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Dimension-Free Rank Lifting from Random Hyperplane Arrangements — arXiv:2609.39855 — arXiv (en), https://arxiv.org/abs/2609.39855 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_dbbc8a13f81f (Metraipolis archive dc_dbbc8a13f81f, sha256 fa2bcab13026b7ef7a58a4b7f6cdcbb1ae19b582bf35d5db8db39ac2a08dc298)

@misc{dcdbbc8a13f81f,
  title = {Dimension-Free Rank Lifting from Random Hyperplane Arrangements — arXiv:2609.39855},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2609.39855},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_dbbc8a13f81f; sha256 fa2bcab13026b7ef7a58a4b7f6cdcbb1ae19b582bf35d5db8db39ac2a08dc298},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2609.39855. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

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 $\theta$, we consider the random feature matrix $\sigma(XR)$, where $R$ is standard Gaussian. For positively homogeneous nonpolynomial activations, which include sign, Heaviside, ReLU, and ReLU powers among others, we prove that $$n \gtrsim \frac{1}{\theta}\max\left\{m,\log\left(\frac{1}{\delta}\right)\right\}$$ neurons suffice for $\sigma(XR)$ to have full row rank $m$ with probability at least $1-\delta$. 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 $\Omega(\theta)$, 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 $$n \gtrsim C^q \frac{m}{\theta^{2q+1}} \log^{2q+\frac{1}{2}}\left(\frac{m}{\theta}\right) \log\left(\frac{m}{\delta}\right),$$ where $q$ is the degree of the activation and $C > 0$ is some universal constant.

Tema degli esploratori: agenti e intelligenza artificiale (retro-catalogo arXiv: annunci del 2026-10-01 in cs.LG).
Categorie arXiv: Machine Learning (cs.LG) ; Probability (math.PR).
Pagina del lavoro: https://arxiv.org/abs/2609.39855 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-07T13:18:10.477Z dalla pagina https://arxiv.org/catchup/cs.LG/2026-10-01?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).