Rapid information (energy) propagation in deep feature extractors is crucial to balance computational complexity versus expressiveness as a representation of the input. We prove an upper bound for the speed of energy propagation in a unified framework that covers different neural network models, both over Euclidean and non-Euclidean domains. Additional structural information about the signal domain can be used to explicitly determine or improve the rate of decay. To illustrate this, we show global exponential energy decay for a range of 1) feature extractors with discrete-domain input signals, and 2) convolutional neural networks (CNNs) via scattering over locally compact abelian (LCA) groups.

翻译：深度特征提取器中信息（能量）的快速传播对于平衡计算复杂度与作为输入表征的表达能力至关重要。我们在一个统一框架下证明了能量传播速度的上界，该框架涵盖了欧几里得域和非欧几里得域上的不同神经网络模型。信号域的额外结构信息可用于显式确定或改进衰减速率。为了说明这一点，我们展示了一系列模型的全局指数能量衰减：1）具有离散域输入信号的特征提取器；2）通过局部紧阿贝尔（LCA）群上的散射实现的卷积神经网络（CNNs）。