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.
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