On decision regions of narrow deep neural networks
This addresses theoretical properties of narrow neural networks, providing insights into their decision boundaries, but is incremental as it builds on existing results.
The paper proves that for neural networks with width less than or equal to the input dimension, all connected components of decision regions are unbounded, applicable to continuous monotonic and ReLU activations, and complements prior work on approximation and connectivity.
We show that for neural network functions that have width less or equal to the input dimension all connected components of decision regions are unbounded. The result holds for continuous and strictly monotonic activation functions as well as for the ReLU activation function. This complements recent results on approximation capabilities by [Hanin 2017 Approximating] and connectivity of decision regions by [Nguyen 2018 Neural] for such narrow neural networks. Our results are illustrated by means of numerical experiments.