Title: $k$-Mixup Regularization for Deep Learning via Optimal Transport. Abstract: Mixup is a popular regularization technique for training deep neural networks that can improve generalization  and increase adversarial robustness. It perturbs input training data in the direction of other randomly-chosen instances in the training set. To better leverage the structure of the data, we extend mixup to $k$-mixup by perturbing $k$-batches of training points in the direction of other $k$-batches using displacement interpolation, i.e. interpolation under the Wasserstein metric. We demonstrate theoretically and in simulations that $k$-mixup preserves cluster and manifold structures, and we extend theory studying the efficacy of standard mixup to the $k$-mixup case. Our empirical results show that training with $k$-mixup further improves generalization and robustness across several network architectures and benchmark datasets of differing modalities.