Mathematics faculty and graduate students are invited to attend the PhD Defense of Christina Cole.

Advisor: Dr. Michael Kirby Co-Advisor: Margaret Cheney

Committee: Dr. Chris Peterson, Dr. Bruce Draper

Title: Cracking Open the Black Box: A Geometric and Topological Analysis of Neural Networks

Abstract: Machine learning is a field that has exploded in recent years in terms of publications and commercial consumption. Despite their increasing prevalence in performing high-risk tasks, machine learning algorithms have outpaced our understanding of them. In this work, we reduce neural networks to their scaffolding defined by polyhedral decompositions. With these decompositions explicitly defined for low-dimensional examples, we utilize novel visualization techniques to build a geometric and topological understanding of them. From there, we develop methods of implicitly accessing neural networks’ polyhedral skeletons, which provide substantial computational and memory savings compared to those requiring explicit access. While much of the related work using neural network polyhedral decompositions is limited to toy models and datasets, the savings provided by our method allow us to use state-of-the-art neural networks and datasets in our analyses. Our experiments alone demonstrate the viability of a polyhedral view of neural networks and our results show its usefulness. More specifically, we show that the geometry that a polyhedral decomposition imposes on its neural network’s domain contains signals that distinguish between original and adversarial images. We conclude our work with suggested future directions. Therefore, we (1) contribute toward closing the gap between our use of neural networks and our understanding of them through geometric and topological analyses and (2) outline avenues for extensions upon this work.