We show that deep networks are better than shallow networks at approximating
functions that can be expressed as a composition of functions described by a
directed acyclic graph, because the deep networks can be designed to have the
same compositional structure, while a shallow network cannot exploit this
knowledge. Thus, the blessing of compositionality mitigates the curse of
dimensionality. On the other hand, a theorem called good propagation of errors
allows to `lift' theorems about shallow networks to those about deep networks
with an appropriate choice of norms, smoothness, etc. We illustrate this in
three contexts where each channel in the deep network calculates a spherical
polynomial, a non-smooth ReLU network, or another zonal function network
related closely with the ReLU network.

利用聚合函数表达的子函数描述构成的有向无环图，深度网络比浅层网络更好地逼近这些函数，因为深度网络可以被设计成具有相同的组合结构，而浅层网络无法利用这一知识，组合性的祝福缓解了维数灾难，而称为良好误差传播的定理允许通过选择适当的范数、平滑度等将有关浅层网络的定理推广到有关深层网络的定理。我们在三个环境中说明了这一点，其中每个通道在深层网络中计算球面多项式、非平滑 ReLU 网络或与 ReLU 网络密切相关的另一种区域函数网络。