Neural Networks, Hypersurfaces, and Radon Transforms
This provides a mathematical framework for understanding neural network phenomena like adversarial examples, which is incremental for researchers in machine learning theory.
The paper tackles the problem of interpreting neural networks by connecting them to integral geometry and Radon transforms, showing that node outputs can be viewed as nonlinear projections along hypersurfaces in the input space.
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.