  Connections between integration along hypersufaces, Radon transforms, and
neural networks are exploited to highlight an integral geometric mathematical
interpretation of neural networks. By analyzing the properties of neural
networks as operators on probability distributions for observed data, we show
that the distribution of outputs for any node in a neural network can be
interpreted as a nonlinear projection along hypersurfaces defined by level
surfaces over the input data space. We utilize these descriptions to provide
new interpretation for phenomena such as nonlinearity, pooling, activation
functions, and adversarial examples in neural network-based learning problems.
