用非局部方法证明深度ReLU网络可高效逼近光滑函数，避免维度灾难。
Nonlocal techniques for the analysis of deep ReLU neural network approximations
- 基于分段线性函数构建非局部逼近框架
- 在0<s<1时，对Sobolev与Barron类函数实现无维数诅咒逼近
- 仅需函数值即可逼近，适合高维数据建模
最近，Daubechies、DeVore、Foucart、Hanin和Petrova引入了一组分段线性函数，这些函数能被带ReLU激活函数的神经网络轻松实现，并构成$L_2([0,1])$上的Riesz基。本文将该结果推广至多变量情形，证明该系统在光滑度$0 < s < 1$下也构成Sobolev空间$W^s([0,1]^d)$和Barron类${b B}^s([0,1]^d)$的Riesz基。我们利用这一性质重新证明了近期关于深度神经网络逼近这些函数类的结果。所提出的证明方法不依赖局部逼近，能够追踪隐含常数，并表明可避免维度诅咒。此外，我们还研究了仅已知函数值时，如何用人工神经网络逼近Sobolev和Barron函数。
原文摘要 · Abstract (English)
Recently, Daubechies, DeVore, Foucart, Hanin, and Petrova introduced a system of piece-wise linear functions, which can be easily reproduced by artificial neural networks with the ReLU activation function and which form a Riesz basis of $L_2([0,1])$. This work was generalized by two of the authors to the multivariate setting. We show that this system serves as a Riesz basis also for Sobolev spaces $W^s([0,1]^d)$ and Barron classes ${\mathbb B}^s([0,1]^d)$ with smoothness $0<s<1$. We apply this fact to re-prove some recent results on the approximation of functions from these classes by deep neural networks. Our proof method avoids using local approximations and allows us to track also the implicit constants as well as to show that we can avoid the curse of dimension. Moreover, we also study how well one can approximate Sobolev and Barron functions by ANNs if only function values are known.
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