This thesis aims to contribute to the field of neural networks by improving upon the performance of a state-of-the-art regularization scheme called MixUp, and by contributing to the conceptual understanding of MixUp. MixUp is a data augmentation scheme in which pairs of training samples and their corresponding labels are mixed using linear coefficients. Without label mixing, MixUp becomes a more conventional scheme: input samples are moved but their original labels are retained. Because samples are preferentially moved in the direction of other classes we refer to this method as directional adversarial training, or DAT. We show that under two mild conditions, MixUp asymptotically convergences to a subset of DAT. We define untied MixUp (UMixUp), a superset of MixUp wherein training labels are mixed with different linear coefficients to those of their corresponding samples. We show that under the same mild conditions, untied MixUp converges to the entire class of DAT schemes. Motivated by the understanding that UMixUp is both a generalization of MixUp and a scheme possessing adversarial-training properties, we experiment with different datasets and loss functions to show that UMixUp provides improves performance over MixUp. In short, we present a novel interpretation of MixUp as belonging to a class highly analogous to adversarial training, and on this basis we introduce a simple generalization which outperforms MixUp.