发表机构
Peking University; Shenzhen Loop Area Institute; King Abdullah University of Science and Technology（北京大学; 深圳河套学院; 阿卜杜拉国王科技大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究针对单位球面上的解析激活线性化浅层网络，推导超球系数的渐近公式，建立尖锐Sobolev速率逼近定理，并给出系数控制，连接谱分析与逼近理论。
AI 中文摘要
我们研究了在单位球面 $\mathbb S^d$ 上具有解析激活函数的线性化浅层网络的谱行为和逼近性质。受早期关于解析带状函数的球面逼近结果 \cite{mhaskar1999approximation} 的启发，我们首先推导出一个渐近公式，该公式以留数显式表示，用于实解析函数的超球系数，该函数的相关奇点是虚轴上的一对共轭简单极点。特别地，该公式在相关的高次子序列上给出了匹配的下界，并确定了指数衰减尺度和符号振荡。它直接适用于这里考虑的 $\tanh$、逻辑斯谛sigmoid和有理激活函数，而 $\arctan$ 的相应估计通过微分获得。对于此类激活函数和拟均匀点 $\{\theta_j^\ast\}_{j=1}^n \subset \mathbb S^d$，满足适当谱相容条件的函数 $f \in \mathcal H^r(\mathbb S^d)$ 可以用线性组合 $$f_n(\eta) = \sum_{j=1}^n a_j \sigma(\theta_j^\ast \cdot \eta)$$ 逼近，并具有尖锐的Sobolev速率 $$\\|f - f_n\\|_{\mathcal H^s(\mathbb S^d)} \lesssim n^{-\frac{r-s}{d}} \\|f\\|_{\mathcal H^r(\mathbb S^d)}.$$ 该逼近定理在左侧放置 $\mathcal H^s$ 范数，同时提供了系数的显式归一化 $\ell^2$ 界。对于谱相容的解析目标函数，相同的构造进一步给出了所述的解析收敛速率。因此，该分析将超球系数不为零的度数处的谱渐近和下界与固定解析激活函数的Sobolev范数逼近和系数控制联系起来。
英文摘要
We study the spectral behavior and approximation properties of linearized shallow networks with analytic activation functions on the unit sphere $\mathbb S^d$. 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 $\{θ_j^\ast\}_{j=1}^n \subset \mathbb S^d$, functions $f \in \mathcal H^r(\mathbb S^d)$ satisfying the appropriate spectral compatibility condition can be approximated by a linear combination $$f_n(η) = \sum_{j=1}^n a_j σ(θ_j^\ast \cdot η),$$ with the sharp Sobolev rate $$\|f - f_n\|_{\mathcal H^s(\mathbb S^d)} \lesssim n^{-\frac{r-s}{d}} \|f\|_{\mathcal H^r(\mathbb S^d)}.$$ The approximation theorem places the $\mathcal H^s$ norm on the left-hand side and simultaneously provides an explicit normalized $\ell^2$ 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.
