提出无维度灾难的浅层神经网络逼近方法，适用于无界域加权函数。
Weighted Sobolev Approximation Rates for Neural Networks on Unbounded Domains
- 在加权Sobolev空间中研究浅层网络逼近能力
- 证明无维度灾难的渐近逼近率，支持无界域与衰减权重
- 适合关注高维函数逼近理论的研究者
本文研究浅层神经网络在加权Sobolev空间中对谱Barron空间函数的逼近能力。现有文献已覆盖若干情形，表明浅层网络可无维度灾难地逼近谱Barron空间中的函数，且适用于多种激活函数。然而，现有结果多局限于有界域上的Sobolev空间误差度量。本文拓展至两类新情形：一是有界域结合Muckenhoupt权重，二是允许无界域且权重需衰减。首先建立更一般的加权Fourier-Lebesgue空间到加权Sobolev空间的嵌入关系，进而推导出浅层神经网络在无维度灾难下的渐近逼近率。
原文摘要 · Abstract (English)
In this work, we consider the approximation capabilities of shallow neural networks in weighted Sobolev spaces for functions in the spectral Barron space. The existing literature already covers several cases, in which the spectral Barron space can be approximated well, i.e., without curse of dimensionality, by shallow networks and several different classes of activation function. The limitations of the existing results are mostly on the error measures that were considered, in which the results are restricted to Sobolev spaces over a bounded domain. We will here treat two cases that extend upon the existing results. Namely, we treat the case with bounded domain and Muckenhoupt weights and the case, where the domain is allowed to be unbounded and the weights are required to decay. We first present embedding results for the more general weighted Fourier-Lebesgue spaces in the weighted Sobolev spaces and then we establish asymptotic approximation rates for shallow neural networks that come without curse of dimensionality.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。