Sharp Approximation on the Sphere by Linearized Shallow Networks with Analytic Activations

Abstract

We study the spectral behavior and approximation properties of linearized shallow networks with analytic activation functions on the unit sphere 𝕊^d. Motivated by earlier spherical approximation results for analytic zonal functions \cite{mhaskar1999approximation}, we first derive an asymptotic formula, expressed explicitly in terms of the residues, for the ultraspherical coefficients of a real-analytic function whose relevant singularities are a conjugate pair of simple poles on the imaginary axis. In particular, the formula gives a matching lower bound along the relevant high-degree subsequences and identifies the exponential decay scale and sign oscillation. It applies directly to tanh, the logistic sigmoid, and the rational activations considered here, while the corresponding estimate for arctan is obtained through differentiation. For such activation functions and quasi-uniform points {θ_j^∗}_j=1ⁿ ⊂ 𝕊^d, functions f ∈ ℋ^r(𝕊^d) satisfying the appropriate spectral compatibility condition can be approximated by a linear combination f_n(η) = ∑_j=1ⁿ a_j σ(θ_j^∗ · η), with the sharp Sobolev rate |f − f_n|_{ℋ^s(𝕊^d)} ≲ n^−(r−s)/d |f|_{ℋ^r(𝕊^d)}. The approximation theorem places the ℋ^s norm on the left-hand side and simultaneously provides an explicit normalized ℓ² bound on the coefficients. For spectrally compatible analytic target functions, the same construction further gives the stated analytic convergence rate. Thus, the analysis connects spectral asymptotics and lower bounds at the degrees where the ultraspherical coefficients do not vanish with Sobolev-norm approximation and coefficient control for fixed analytic activation functions.

Fully Deterministic Shallow Neural Networks with Localized Activations. Part I: Construction and Optimal Sobolev Approximation
Ran Bi, Weibing Deng
arXiv7 Oct 2026PreprintarXiv licenceFull textNumerical analysis & scientific computingarXiv
Fully Deterministic Shallow Neural Networks with Localized Activations. Part II: Condition Number Estimates
Ran Bi, Weibing Deng
arXiv7 Oct 2026PreprintarXiv licenceFull textNumerical analysis & scientific computingarXiv
Parameter Bounds and Inverse Approximation for Linearized ReLU^k Networks on the Sphere
Yuqing Liu, Tong Mao, Jonathan W. Siegel +1
arXiv1 Nov 2026PreprintCC BY 4.0Full textNumerical analysis & scientific computingarXiv
Diffusion Score Estimation with Shallow Neural Networks: Regularity, Approximation, and Finite-Sample Rates
Fan Liu, Yahong Yang, Wei Zhu
arXiv19 Oct 2026PreprintarXiv licenceFull textStatisticsarXiv
Two-Dimensional Shallow Water Linearized Moment Equations: Hyperbolicity and Well-Balanced Schemes
Shiping Zhou, Juntao Huang, Andrew Christlieb
arXiv23 Sep 2026PreprintarXiv licenceFull textNumerical analysis & scientific computingarXiv

Cite

BibTeX RIS

BibTeX

@misc{li2026sharp,
  title = {{Sharp Approximation on the Sphere by Linearized Shallow Networks with Analytic Activations}},
  author = {Jia Li and Tong Mao and Jinchao Xu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.26276},
  eprint = {2609.26276},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.26276},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Sharp Approximation on the Sphere by Linearized Shallow Networks with Analytic Activations
AU  - Jia Li
AU  - Tong Mao
AU  - Jinchao Xu
PY  - 2026
DA  - 2026/09/01
PB  - arXiv
DO  - 10.48550/arxiv.2609.26276
UR  - https://arxiv.org/abs/2609.26276
AB  - We study the spectral behavior and approximation properties of linearized shallow networks with analytic activation functions on the unit sphere 𝕊^d. Motivated by earlier spherical approximation results for analytic zonal functions \cite{mhaskar1999approximation}, we first derive an asymptotic formula, expressed explicitly in terms of the residues, for the ultraspherical coefficients of a real-analytic function whose relevant singularities are a conjugate pair of simple poles on the imaginary axis. In particular, the formula gives a matching lower bound along the relevant high-degree subsequences and identifies the exponential decay scale and sign oscillation. It applies directly to tanh, the logistic sigmoid, and the rational activations considered here, while the corresponding estimate for arctan is obtained through differentiation. For such activation functions and quasi-uniform points {θ_j^∗}_j=1^n ⊂ 𝕊^d, functions f ∈ ℋ^r(𝕊^d) satisfying the appropriate spectral compatibility condition can be approximated by a linear combination f_n(η) = ∑_j=1^n a_j σ(θ_j^∗ · η), with the sharp Sobolev rate |f − f_n|_ℋ^s(𝕊^d) ≲ n^−(r−s)/d |f|_ℋ^r(𝕊^d). The approximation theorem places the ℋ^s norm on the left-hand side and simultaneously provides an explicit normalized ℓ^2 bound on the coefficients. For spectrally compatible analytic target functions, the same construction further gives the stated analytic convergence rate. Thus, the analysis connects spectral asymptotics and lower bounds at the degrees where the ultraspherical coefficients do not vanish with Sobolev-norm approximation and coefficient control for fixed analytic activation functions.
ER  -

Abstract

Related

Cite