Parameter Bounds and Inverse Approximation for Linearized ReLUk Networks on the Sphere
Abstract
We study direct and inverse approximation for linearized shallow ReLUk networks on 𝕊d with quasi-uniform hidden parameters. For 0<α<(d+2k+1)/2d, under the coefficient growth condition (n∑j=1naj,n2)1/2≤ nαM, we prove that approximation at the order n−(d+2k+1)/2d+α is equivalent to membership in the parity-compatible Besov space ℬ2,∞,*(d+2k+1)/2−αd(𝕊d). Mere convergence under the same coefficient budget does not imply this regularity. For unconstrained approximation along a nested parameter sequence, a single ridge function gives a counterexample when α<1/2, whereas for α>1/2 the endpoint Besov inverse conclusion follows from a Bernstein inequality under antipodal quasi-uniformity. The inverse conclusion is also recovered when the prescribed rate holds for two quasi-uniform parameter sequences whose union is antipodally separated.
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BibTeX
@misc{liu2026parameter,
title = {{Parameter Bounds and Inverse Approximation for Linearized ReLU$^k$ Networks on the Sphere}},
author = {Yuqing Liu and Tong Mao and Jonathan W. Siegel and Jinchao Xu},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.00656},
eprint = {2611.00656},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.00656},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Parameter Bounds and Inverse Approximation for Linearized ReLU^k Networks on the Sphere
AU - Yuqing Liu
AU - Tong Mao
AU - Jonathan W. Siegel
AU - Jinchao Xu
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.00656
UR - https://arxiv.org/abs/2611.00656
AB - We study direct and inverse approximation for linearized shallow ReLU^k networks on 𝕊^d with quasi-uniform hidden parameters. For 0<α<(d+2k+1)/2d, under the coefficient growth condition (n∑_j=1^na_j,n^2)^1/2≤ n^αM, we prove that approximation at the order n^−(d+2k+1)/2d+α is equivalent to membership in the parity-compatible Besov space ℬ_2,∞,*^(d+2k+1)/2−αd(𝕊^d). Mere convergence under the same coefficient budget does not imply this regularity. For unconstrained approximation along a nested parameter sequence, a single ridge function gives a counterexample when α<1/2, whereas for α>1/2 the endpoint Besov inverse conclusion follows from a Bernstein inequality under antipodal quasi-uniformity. The inverse conclusion is also recovered when the prescribed rate holds for two quasi-uniform parameter sequences whose union is antipodally separated.
ER -