arXiv：cs.LG· Yuwen Li, Guozhi Zhang·· 6 小时前AI 评分29
浅层神经网络在混合 Sobolev 空间中的逼近研究
Shallow neural network approximation in mixed Sobolev spaces
AI 导读
研究用 n 个神经元、一般激活函数的浅层神经网络对混合 Sobolev 空间进行最优 L2 逼近，提出与激活函数无关的 Fourier-block 原理：若激活函数单变量逼近阶为 ρ，则混合光滑度 α 的目标函数全局逼近率为 min{α,ρ}（至多对数因子）。
正文
Abstract: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 $\rho$ in the sense of the Fourier-block property, then the global approximation rate has algebraic order $\min\{\alpha,\rho\}$ for target functions of mixed smoothness $\alpha$, 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 $\min\{\alpha,k+1\}$ as the optimal algebraic approximation exponent in any dimension, up to logarithmic factors in the upper bound. The framework also yields the exponent $\min\{\alpha,k+1\}$ for cardinal B-splines and soft-$\mathrm{ReLU}^k$, and the full mixed-smoothness exponent $\alpha$ for ELU and cosine activations, again up to logarithmic~factors.
| Comments: | 40 pages, 2 figures |
| Subjects: | Numerical Analysis (math.NA); Machine Learning (cs.LG) |
| Cite as: | arXiv:2609.05263 [math.NA] |
| (or arXiv:2609.05263v2 [math.NA] for this version) | |
| https://doi.org/10.48550/arXiv.2609.05263 arXiv-issued DOI via DataCite |
Submission history
From: Guozhi Zhang [view email]
[v1]
Fri, 4 Sep 2026 15:23:34 UTC (39 KB)
[v2]
Wed, 7 Oct 2026 14:20:33 UTC (41 KB)
来源：arXiv：cs.LG · arxiv.org