Certifying Robustness via Topological Representations

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.