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Data augmentation (DA), an effective regularization technique, generates training samples to enhance the diversity of data and the richness of label information for training modern deep learning models. mixup, a popular recent DA method, augments training datasets with convex combinations of original samples pairs, but can generate undesirable samples, with data being sampled off the manifold and with incorrect labels. In this work, we propose ζ-mixup, a generalization of mixup with provably and demonstrably desirable properties that allows for convex combinations of $N \geq 2$ samples, thus leading to more realistic and diverse outputs that incorporate information from $N$ original samples using a $p$-series interpolant. We show that, compared to mixup, ζ-mixup better preserves the intrinsic dimensionality of the original datasets, a desirable property for training generalizable models, and is at least as fast as mixup. Evaluation on several natural and medical image datasets shows that ζ-mixup outperforms mixup, CutMix, and traditional DA methods.
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