  Deep neural networks (DNNs) are vulnerable to small adversarial perturbations
of the inputs, posing a significant challenge to their reliability and
robustness. Empirical methods such as adversarial training can defend against
particular attacks but remain vulnerable to more powerful attacks.
Alternatively, Lipschitz networks provide certified robustness to unseen
perturbations but lack sufficient expressive power. To harness the advantages
of both approaches, we design a novel two-step Optimal Transport induced
Adversarial Defense (OTAD) model that can fit the training data accurately
while preserving the local Lipschitz continuity. First, we train a DNN with a
regularizer derived from optimal transport theory, yielding a discrete optimal
transport map linking data to its features. By leveraging the map's inherent
regularity, we interpolate the map by solving the convex integration problem
(CIP) to guarantee the local Lipschitz property. OTAD is extensible to diverse
architectures of ResNet and Transformer, making it suitable for complex data.
For efficient computation, the CIP can be solved through training neural
networks. OTAD opens a novel avenue for developing reliable and secure deep
learning systems through the regularity of optimal transport maps. Empirical
results demonstrate that OTAD can outperform other robust models on diverse
datasets.
