# When Can Neural Networks Learn Connected Decision Regions?

## TL;DR

This paper investigates the conditions under which neural networks can learn connected decision regions, extending previous results to a broader class of activation functions and providing a deeper theoretical understanding.

## Contribution

It advances the theory by establishing necessary and sufficient conditions for connected decision regions across various activation functions, beyond pyramidal structures.

## Key findings

- Connected decision regions depend on network architecture and activation functions.
- Broader class of activation functions can produce connected decision regions under certain conditions.
- Theoretical framework applies to widely used activation functions like ReLU, sigmoid, and tanh.

## Abstract

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.