This article investigates whether partial data augmentation can achieve the same statistical benefits as full augmentation by developing a framework using Fourier analysis and representation theory of finite groups.
- Partial data augmentation based on a randomly sampled subset of group elements achieves the same minimax rates as full augmentation, up to an approximation error that vanishes as the subset size increases.
- The study proves an impossibility result stating that enforcing exact invariance requires averaging over the entire group and cannot be achieved by any strict subset when the hypothesis space is sufficiently expressive.
- These findings provide a theoretical explanation for why partial augmentation retains statistical benefits despite enforcing symmetry only approximately.
The results offer a unified perspective on full and partial data augmentation, addressing whether statistically optimal learning under general group invariances can be achieved using computationally scalable methods.