# Characterizing the Shape of Activation Space in Deep Neural Networks

## TL;DR

This paper introduces a topological method using persistent homology to analyze neural network activation spaces, revealing new insights into their structure and the nature of adversarial examples.

## Contribution

The paper presents a novel topological approach to interpret neural network representations, providing a new perspective on adversarial examples and activation structures.

## Key findings

- Adversarial examples alter dominant activation structures rather than target semantic features.
- Neural network class representations are sparsely distributed in input space.
- Topological analysis offers unique insights into distributed neural representations.

## Abstract

The representations learned by deep neural networks are difficult to interpret in part due to their large parameter space and the complexities introduced by their multi-layer structure. We introduce a method for computing persistent homology over the graphical activation structure of neural networks, which provides access to the task-relevant substructures activated throughout the network for a given input. This topological perspective provides unique insights into the distributed representations encoded by neural networks in terms of the shape of their activation structures. We demonstrate the value of this approach by showing an alternative explanation for the existence of adversarial examples. By studying the topology of network activations across multiple architectures and datasets, we find that adversarial perturbations do not add activations that target the semantic structure of the adversarial class as previously hypothesized. Rather, adversarial examples are explainable as alterations to the dominant activation structures induced by the original image, suggesting the class representations learned by deep networks are problematically sparse on the input space.