When Can Neural Networks Learn Connected Decision Regions?
This work addresses a theoretical problem in understanding neural network behavior, with implications for robustness and adversarial manipulation, but it is incremental as it builds on prior results.
The paper advances the theory of when neural networks produce connected decision regions, extending previous results to a wider class of activation functions including ReLU, sigmoid, tanh, softplus, and exponential linear functions.
Previous work has questioned the conditions under which the decision regions of a neural network are connected and further showed the implications of the corresponding theory to the problem of adversarial manipulation of classifiers. It has been proven that for a class of activation functions including leaky ReLU, neural networks having a pyramidal structure, that is no layer has more hidden units than the input dimension, produce necessarily connected decision regions. In this paper, we advance this important result by further developing the sufficient and necessary conditions under which the decision regions of a neural network are connected. We then apply our framework to overcome the limits of existing work and further study the capacity to learn connected regions of neural networks for a much wider class of activation functions including those widely used, namely ReLU, sigmoid, tanh, softlus, and exponential linear function.