Certifying Robustness via Topological Representations

Exploring the shape of data spaces is providing new insights in data analysis and deep learning tasks within a variety of application domains. A common approach in Topological Data Analysis to extract multi-scale intrinsic geometric properties of data is persistent homology. This method 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 underexplored. 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 $ϵ$-robustness for samples in a dataset, which we demonstrate on the ORBIT5K data set representing the orbits of a discrete dynamical system.