关于多元神经网络逼近的误差界划分
On Sharpness of Error Bounds for Multivariate Neural Network Approximation
摘要：
单隐层前馈神经网络可以表示作为脊函数之和的多变量函数。这些脊函数通过激活函数和可定制的权重来定义。本文讨论了这类岭函数和的最佳非线性逼近。误差界的光滑模。然而，主要的重点是证明边界是最好的可能。为此，反例构造的非线性，定量扩展的一致有界性原则。他们表现出锋利度相对于Lipschitz类的逻辑激活函数和某些分段多项式激活函数。该文件是基于单变量的结果（戈培尔，圣：单隐层前馈神经网络单变量逼近的误差界的尖锐性。结果数学75（3），2020，文章109，https：//rdcu.be/b5mKH）。
英文摘要：
Single hidden layer feedforward neural networks can represent multivariate functions that are sums of ridge functions. These ridge functions are defined via an activation function and customizable weights. The paper deals with best non-linear approximation by such sums of ridge functions. Error bounds are presented in terms of moduli of smoothness. The main focus, however, is to prove that the bounds are best possible. To this end, counterexamples are constructed with a non-linear, quantitative extension of the uniform boundedness principle. They show sharpness with respect to Lipschitz classes for the logistic activation function and for certain piecewise polynomial activation functions. The paper is based on univariate results in (Goebbels, St.: On sharpness of error bounds for univariate approximation by single hidden layer feedforward neural networks. Results Math 75 (3), 2020, article 109, https://rdcu.be/b5mKH).
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