  We propose regularization strategies for learning discriminative models that
are robust to in-class variations of the input data. We use the Wasserstein-2
geometry to capture semantically meaningful neighborhoods in the space of
images, and define a corresponding input-dependent additive noise data
augmentation model. Expanding and integrating the augmented loss yields an
effective Tikhonov-type Wasserstein diffusion smoothness regularizer. This
approach allows us to apply high levels of regularization and train functions
that have low variability within classes but remain flexible across classes. We
provide efficient methods for computing the regularizer at a negligible cost in
comparison to training with adversarial data augmentation. Initial experiments
demonstrate improvements in generalization performance under adversarial
perturbations and also large in-class variations of the input data.
