发表机构
Zhejiang University（浙江大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究建立浅层神经网络对混合Sobolev空间的逼近理论，提出傅里叶块原理，确定ReLUᵏ等多种激活函数的最优逼近指数，为相关逼近问题提供了理论框架。
AI 中文摘要
我们研究具有n个神经元和一般激活函数的浅层神经网络对混合Sobolev空间的最佳L₂逼近。首先建立与激活函数无关的傅里叶块原理：若某激活函数在傅里叶块性质意义下具有单变量逼近阶ρ，则对于混合光滑度为α的目标函数，全局逼近速率具有代数阶min{α,ρ}，且存在显式对数因子。为验证具体激活函数的该性质，我们引入结构化单变量逼近条件，该条件可导出带显式参数的傅里叶块性质。对于ReLUᵏ，匹配的代数下界确定了任意维度下最优代数逼近指数为min{α,k+1}，且上界中存在对数因子；该框架对基数B样条和soft-ReLUᵏ也给出指数min{α,k+1}，对ELU和余弦激活函数则给出完整混合光滑度指数α，同样带有对数因子。
英文摘要
We investigate the best $L_2$ approximation of mixed Sobolev spaces by shallow neural networks with $n$ neurons and general activation functions. We first establish an activation-independent Fourier-block principle: if an activation has univariate approximation order $ρ$ in the sense of the Fourier-block property, then the global approximation rate has algebraic order $\min\{α,ρ\}$ for target functions of mixed smoothness $α$, up to explicit logarithmic factors. To verify this property for concrete activations, we introduce a structured univariate approximation condition that implies the Fourier-block property with explicit parameters. For $\mathrm{ReLU}^k$, a matching algebraic lower bound identifies $\min\{α,k+1\}$ as the optimal algebraic approximation exponent in any dimension, up to logarithmic factors in the upper bound. The framework also yields the exponent $\min\{α,k+1\}$ for cardinal B-splines and soft-$\mathrm{ReLU}^k$, and the full mixed-smoothness exponent $α$ for ELU and cosine activations, again up to logarithmic~factors.
Comments40 pages, 2 figures
