A Group-Theoretic Framework for Data Augmentation
This provides a theoretical foundation for data augmentation, which is widely used but poorly understood, potentially improving methods in fields like cryo-EM.
The authors tackled the lack of a mathematical framework to explain data augmentation's benefits in deep learning, showing it is equivalent to group averaging that reduces variance and applying this to empirical risk minimization and specific models like neural networks.
Data augmentation is a widely used trick when training deep neural networks: in addition to the original data, properly transformed data are also added to the training set. However, to the best of our knowledge, a clear mathematical framework to explain the performance benefits of data augmentation is not available. In this paper, we develop such a theoretical framework. We show data augmentation is equivalent to an averaging operation over the orbits of a certain group that keeps the data distribution approximately invariant. We prove that it leads to variance reduction. We study empirical risk minimization, and the examples of exponential families, linear regression, and certain two-layer neural networks. We also discuss how data augmentation could be used in problems with symmetry where other approaches are prevalent, such as in cryo-electron microscopy (cryo-EM).