  Deep neural networks, particularly in vision tasks, are notably susceptible
to adversarial perturbations. To overcome this challenge, developing a robust
classifier is crucial. In light of the recent advancements in the robustness of
classifiers, we delve deep into the intricacies of adversarial training and
Jacobian regularization, two pivotal defenses. Our work is the first carefully
analyzes and characterizes these two schools of approaches, both theoretically
and empirically, to demonstrate how each approach impacts the robust learning
of a classifier. Next, we propose our novel Optimal Transport with Jacobian
regularization method, dubbed OTJR, bridging the input Jacobian regularization
with the a output representation alignment by leveraging the optimal transport
theory. In particular, we employ the Sliced Wasserstein distance that can
efficiently push the adversarial samples' representations closer to those of
clean samples, regardless of the number of classes within the dataset. The SW
distance provides the adversarial samples' movement directions, which are much
more informative and powerful for the Jacobian regularization. Our empirical
evaluations set a new standard in the domain, with our method achieving
commendable accuracies of 52.57% on CIFAR-10 and 28.3% on CIFAR-100 datasets
under the AutoAttack. Further validating our model's practicality, we conducted
real-world tests by subjecting internet-sourced images to online adversarial
attacks. These demonstrations highlight our model's capability to counteract
sophisticated adversarial perturbations, affirming its significance and
applicability in real-world scenarios.
