发表机构
Nanjing University（南京大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了具有高斯和tanh差分激活的确定性浅层网络在Sobolev Gram矩阵上的正定性和谱条件数上界，并给出匹配下界，实验验证了条件数增长与归一化效果。
AI 中文摘要
我们证明了具有高斯和tanh差分激活函数的确定性浅层网络的连续Sobolev Gram矩阵的正定性和定量条件数界。所规定的特征族包括第一部分中的最优Sobolev逼近构造。在$\R^d$（$d\ge2$）中每个固定的非空有界Lipschitz域上，以及每个固定的整数$m\ge0$，在完全$H^m$内积下的原始和列归一化Gram矩阵的谱条件数对于足够大的$N$以$\exp(CN^{2/d}/\log(N+2))$为界，其中$N$是特征数量，$C>0$与$N$无关。上界对中点和Gauss-Legendre偏移成立。对于中点偏移和合适的角序列，在固定椭球上的匹配下界确立了该指数在所述域类上对原始高斯矩阵以及原始和归一化tanh差分矩阵的尖锐性。证明结合了隔离方向组的齐次微分算子与一维系数恢复估计。在正方形上的二维$L^2$实验说明了条件数增长和归一化效应。
英文摘要
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 $\R^d$, $d\ge2$, and for every fixed integer $m\ge0$, 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.
