Sharp approximation rates for shallow ReLU neuralnetworks on critical Besov classes
Abstract
Let 𝔻 be the normalized ridge dictionary generated by ReLUk on a bounded Lipschitz domain Ω⊂ℝd. We determine the sharp algebraic rate of finite n-term approximation from 𝔻 when the outer ℓ1 coefficient budget is independent of n. More precisely, let d≥3, k∈ℕ+, 0≤ m≤ k, 0<p<1, and 0<q≤1. For the unit ball of the critical Besov space Bp,qk+d/p(Ω), measured in Hm(Ω), the optimal algebraic exponent is min{(k−m+d/2)/(d−1), (k−m+d/2+1/p−1/2)/d}. We prove two-sided estimates in the three regimes determined by the two branches of this minimum and give the corresponding logarithmic factors at and beyond the transition. The estimates coincide without logarithmic loss in the strict angular regime and at the transition when q≤[1/2+(k−m+d/2)/(d−1)]−1. The upper estimate combines critical wavelet sparsity, localized Fourier--Radon representations, and a stable allocation of directions and biases. The lower estimates arise from two distinct obstructions, namely radial ridge approximation on the direction sphere and Gevrey localization in the joint direction--bias space. The critical smoothness is exactly the endpoint at which Besov regularity provides a uniform shallow-network variation bound without additional scale decay. The resulting representation exponent is strictly larger than the degree-limited exponent for fixed-degree isotropic finite elements under a parameter-count comparison. This last statement concerns best approximation, not training or computational complexity.
Related
- Sharp Approximation on the Sphere by Linearized Shallow Networks with Analytic Activations
- Shallow neural network approximation in mixed Sobolev spaces
- Boundary Geometry Determines Optimal Shallow ReLU Approximation under Weighted Coefficient Constraints
- Sharp embeddings between quasi-Banach Besov spaces and shallow ReLU variation spaces
- Fully Deterministic Shallow Neural Networks with Localized Activations. Part I: Construction and Optimal Sobolev Approximation
BibTeX
@misc{wang2026sharp,
title = {{Sharp approximation rates for shallow ReLU neuralnetworks on critical Besov classes}},
author = {Yupeng Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06418},
eprint = {2609.06418},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06418},
}
RIS
TY - UNPB
TI - Sharp approximation rates for shallow ReLU neuralnetworks on critical Besov classes
AU - Yupeng Wang
PY - 2026
DA - 2026/09/06
PB - arXiv
DO - 10.48550/arxiv.2609.06418
UR - https://arxiv.org/abs/2609.06418
AB - Let 𝔻 be the normalized ridge dictionary generated by ReLU^k on a bounded Lipschitz domain Ω⊂ℝ^d. We determine the sharp algebraic rate of finite n-term approximation from 𝔻 when the outer ℓ^1 coefficient budget is independent of n. More precisely, let d≥3, k∈ℕ_+, 0≤ m≤ k, 0<p<1, and 0<q≤1. For the unit ball of the critical Besov space B_p,q^k+d/p(Ω), measured in H^m(Ω), the optimal algebraic exponent is min{(k−m+d/2)/(d−1), (k−m+d/2+1/p−1/2)/d}. We prove two-sided estimates in the three regimes determined by the two branches of this minimum and give the corresponding logarithmic factors at and beyond the transition. The estimates coincide without logarithmic loss in the strict angular regime and at the transition when q≤[1/2+(k−m+d/2)/(d−1)]^−1. The upper estimate combines critical wavelet sparsity, localized Fourier--Radon representations, and a stable allocation of directions and biases. The lower estimates arise from two distinct obstructions, namely radial ridge approximation on the direction sphere and Gevrey localization in the joint direction--bias space. The critical smoothness is exactly the endpoint at which Besov regularity provides a uniform shallow-network variation bound without additional scale decay. The resulting representation exponent is strictly larger than the degree-limited exponent for fixed-degree isotropic finite elements under a parameter-count comparison. This last statement concerns best approximation, not training or computational complexity.
ER -