研究浅层神经网络在巴龙空间中的逼近能力，揭示激活函数与维度的权衡关系。
Approximation Rates of Shallow Neural Networks: Barron Spaces, Activation Functions and Optimality Analysis
- 分析幂指数型激活函数在巴龙空间中的逼近效果
- 证明ReLU^k在ℓ¹系数约束下无法达到最优逼近率
- 揭示维度越高逼近越难，适合理论研究者参考
本文研究具有幂指数型激活函数的浅层神经网络的逼近性质，重点分析在巴龙函数空间中逼近率随维度和被逼近函数光滑性变化的关系。针对ReLU^k激活函数，证明在ℓ¹-有界系数或不足光滑条件下，无法实现最优逼近率。同时，在不同范数下建立了巴龙空间和索博列夫空间中函数的最优逼近率，验证了维数灾难的存在。研究成果厘清了浅层神经网络逼近能力的边界，为激活函数与网络结构的选择提供了理论依据。
原文摘要 · Abstract (English)
This paper investigates the approximation properties of shallow neural networks with activation functions that are powers of exponential functions. It focuses on the dependence of the approximation rate on the dimension and the smoothness of the function being approximated within the Barron function space. We examine the approximation rates of ReLU$^{k}$ activation functions, proving that the optimal rate cannot be achieved under $\ell^{1}$-bounded coefficients or insufficient smoothness conditions. We also establish optimal approximation rates in various norms for functions in Barron spaces and Sobolev spaces, confirming the curse of dimensionality. Our results clarify the limits of shallow neural networks' approximation capabilities and offer insights into the selection of activation functions and network structures.
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