Certifying Robustness via Topological Representations

Jens Agerberg, Andrea Guidolin, Andrea Martinelli, Pepijn Roos Hoefgeest, David Eklund, Martina Scolamiero

Keywords: topological data analysis, persistent homology, topological machine learning, adversarial machine learning, robustness

TL;DR: We introduce a neural network designed to learn robust and discriminative geometric representations from persistent homology (PH) using an adversarial learning framework, addressing challenges of robustness when integrating PH with neural networks.

Abstract: In machine learning, the ability to obtain representations that capture underlying geometrical and topological structures of data spaces is crucial. A common approach in Topological Data Analysis to extract multi-scale intrinsic geometric properties of data is persistent homology. This methods enjoys theoretical stability results (i.e Lipschitz continuity with respect to appropriate metrics), however the significance of this robustness when persistent homology is used in machine learning is under-explored. We propose a neural network architecture that can learn discriminative geometric representations from persistence with a controllable Lipschitz constant. In adversarial learning, this end-to-end stability can be used to certify $\epsilon$-robustness for samples in a dataset, which we demonstrate on the ORBIT5K data set representing the orbits of a discrete dynamical system.

Submission Number: 27