We establish a scale separation of Kolmogorov width type between subspaces of a given Banach space under the condition that a sequence of linear maps converges much faster on one of the subspaces. The general technique is then applied to show that reproducing kernel Hilbert spaces are poor \(L^{2}\)-approximators for the class of two-layer neural networks in high dimension, and that multi-layer networks with small path norm are poor approximators for certain Lipschitz functions, also in the \(L^{2}\)-topology.

中文翻译：

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机器学习中的Kolmogorov宽度衰减和近似值差：浅层神经网络，随机特征模型和神经正切核

我们在给定Banach空间的子空间之间建立Kolmogorov宽度类型的尺度分离，条件是线性映射序列在一个子空间上收敛得更快。然后应用通用技术表明，对于高维两层神经网络，再现核希尔伯特空间的穷人\（L ^ {2} \） -逼近器，并且具有小路径范数的多层网络某些Lipschitz函数（在\（L ^ {2} \）-拓扑中）的近似值也很差。