Fully Deterministic Shallow Neural Networks with Localized Activations. Part II: Condition Number Estimates
Abstract
We prove positive definiteness and quantitative condition-number bounds for continuous Sobolev Gram matrices of deterministic shallow networks with Gaussian and tanh-difference activations. The prescribed feature families include the optimal Sobolev approximation constructions of Part I. On every fixed nonempty bounded Lipschitz domain in Rd, d≥2, and for every fixed integer m≥0, the raw and column-normalized Gram matrices in the full Hm inner product have spectral condition numbers bounded by exp(CN2/d/log(N+2)) for sufficiently large N, where N is the feature count and C>0 is independent of N. The upper bounds hold for midpoint and Gauss-Legendre offsets. For midpoint offsets and a suitable angular sequence, matching lower bounds on fixed ellipsoids establish sharpness of this exponent over the stated domain class for raw Gaussian matrices and both raw and normalized tanh-difference matrices. The proof combines homogeneous differential operators that isolate direction groups with one-dimensional coefficient-recovery estimates. Two-dimensional L2 experiments on a square illustrate condition-number growth and normalization effects.
Related
- Fully Deterministic Shallow Neural Networks with Localized Activations. Part I: Construction and Optimal Sobolev Approximation
- Radon Measure Representations for Infinite-Width Neural Networks with Singular Activations
- Sharp Approximation on the Sphere by Linearized Shallow Networks with Analytic Activations
- The Two-Dimensional Narrow Escape Problem with Deterministic Initial Condition: Mean Escape Time and Escape location probabilities
- Localized maximum-norm error estimates for the Hellan-Herrmann-Johnson method
BibTeX
@misc{bi2026fully,
title = {{Fully Deterministic Shallow Neural Networks with Localized Activations. Part II: Condition Number Estimates}},
author = {Ran Bi and Weibing Deng},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.09357},
eprint = {2610.09357},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.09357},
}
RIS
TY - UNPB
TI - Fully Deterministic Shallow Neural Networks with Localized Activations. Part II: Condition Number Estimates
AU - Ran Bi
AU - Weibing Deng
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.09357
UR - https://arxiv.org/abs/2610.09357
AB - We prove positive definiteness and quantitative condition-number bounds for continuous Sobolev Gram matrices of deterministic shallow networks with Gaussian and tanh-difference activations. The prescribed feature families include the optimal Sobolev approximation constructions of Part I. On every fixed nonempty bounded Lipschitz domain in R^d, d≥2, and for every fixed integer m≥0, the raw and column-normalized Gram matrices in the full H^m inner product have spectral condition numbers bounded by exp(CN^2/d/log(N+2)) for sufficiently large N, where N is the feature count and C>0 is independent of N. The upper bounds hold for midpoint and Gauss-Legendre offsets. For midpoint offsets and a suitable angular sequence, matching lower bounds on fixed ellipsoids establish sharpness of this exponent over the stated domain class for raw Gaussian matrices and both raw and normalized tanh-difference matrices. The proof combines homogeneous differential operators that isolate direction groups with one-dimensional coefficient-recovery estimates. Two-dimensional L^2 experiments on a square illustrate condition-number growth and normalization effects.
ER -