# Data Augmentation: A Fourier Analysis Perspective

arXiv: https://arxiv.org/abs/2606.24418 · HTML: https://mlsanity.com/paper/2606.24418/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 37 of its 60 references that are in the map.
- First posted on arXiv 23 Jun 2026.

## Builds on (most cited first, 30 of 37)

- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [E(3)-Equivariant Graph Neural Networks for Data-Efficient and Accurate Interatomic Potentials](https://mlsanity.com/paper/2101.03164/) (2021)
- [Frame Averaging for Invariant and Equivariant Network Design](https://mlsanity.com/paper/2110.03336/) (2021)
- [Artificial Intelligence for Science in Quantum, Atomistic, and Continuum Systems](https://mlsanity.com/paper/2307.08423/) (2023)
- [Equivariance with Learned Canonicalization Functions](https://mlsanity.com/paper/2211.06489/) (2022)
- [A Kernel Theory of Modern Data Augmentation](https://mlsanity.com/paper/1803.06084/) (2018)
- [Approximately Equivariant Networks for Imperfectly Symmetric Dynamics](https://mlsanity.com/paper/2201.11969/) (2022)
- [Learning with invariances in random features and kernel models](https://mlsanity.com/paper/2102.13219/) (2021)
- [Automatic Symmetry Discovery with Lie Algebra Convolutional Network](https://mlsanity.com/paper/2109.07103/) (2021)
- [Data Augmentation Approaches in Natural Language Processing: A Survey](https://mlsanity.com/paper/2110.01852/) (2021)
- [Residual Pathway Priors for Soft Equivariance Constraints](https://mlsanity.com/paper/2112.01388/) (2021)
- [Graph Data Augmentation for Graph Machine Learning: A Survey](https://mlsanity.com/paper/2202.08871/) (2022)
- [Approximation-Generalization Trade-offs under (Approximate) Group Equivariance](https://mlsanity.com/paper/2305.17592/) (2023)
- [Learning Partial Equivariances from Data](https://mlsanity.com/paper/2110.10211/) (2021)
- [Equivariant Frames and the Impossibility of Continuous Canonicalization](https://mlsanity.com/paper/2402.16077/) (2024)
- [Generative Adversarial Symmetry Discovery](https://mlsanity.com/paper/2302.00236/) (2023)
- [Relaxing Equivariance Constraints with Non-stationary Continuous Filters](https://mlsanity.com/paper/2204.07178/) (2022)
- [Latent Space Symmetry Discovery](https://mlsanity.com/paper/2310.00105/) (2023)
- [Sample Efficiency of Data Augmentation Consistency Regularization](https://mlsanity.com/paper/2202.12230/) (2022)
- [The Exact Sample Complexity Gain from Invariances for Kernel Regression](https://mlsanity.com/paper/2303.14269/) (2023)
- [A Canonicalization Perspective on Invariant and Equivariant Learning](https://mlsanity.com/paper/2405.18378/) (2024)
- [A Comprehensive Survey of Data Augmentation in Visual Reinforcement Learning](https://mlsanity.com/paper/2210.04561/) (2022)
- [Toward Understanding Generative Data Augmentation](https://mlsanity.com/paper/2305.17476/) (2023)
- [The good, the bad and the ugly sides of data augmentation: An implicit spectral regularization perspective](https://mlsanity.com/paper/2210.05021/) (2022)
- [Lie Algebra Canonicalization: Equivariant Neural Operators under arbitrary Lie Groups](https://mlsanity.com/paper/2410.02698/) (2024)
- [Learning Layer-wise Equivariances Automatically using Gradients](https://mlsanity.com/paper/2310.06131/) (2023)
- [Regularizing Towards Soft Equivariance Under Mixed Symmetries](https://mlsanity.com/paper/2306.00356/) (2023)
- [Symmetry Discovery Beyond Affine Transformations](https://mlsanity.com/paper/2406.03619/) (2024)
- [Understanding the Detrimental Class-level Effects of Data Augmentation](https://mlsanity.com/paper/2401.01764/) (2023)
- [On the hardness of learning under symmetries](https://mlsanity.com/paper/2401.01869/) (2024)

## Built on by (most cited first, 4 of 4)

- [Adaptive Symmetry Discovery for Dynamical System Identification](https://mlsanity.com/paper/2608.08091/) (2026)
- [Sparse Data Augmentation for Optimization with Provable Guarantees](https://mlsanity.com/paper/2609.08133/) (2026)
- [Interpolation Is Not Invariance: Pair Count Is Not Coverage in Transformation Audits](https://mlsanity.com/paper/2609.14870/) (2026)
- [Dimension-Free Scaling Laws for Size Generalization: An Exponential Separation](https://mlsanity.com/paper/2610.19279/) (2026)

## Abstract (arXiv)

Data augmentation is a simple and model-agnostic approach for exploiting known invariances in learning problems. Given a group acting on the input space, one augments the training set with transformed copies of each sample. Because it exploits symmetries without modifying the underlying learning algorithm, data augmentation can be applied broadly across learning methods. However, this universality comes at a computational cost: when the group is large, full group-sized augmentation quickly becomes computationally infeasible. This raises a fundamental question: Can partial data augmentation achieve the same statistical benefits as full augmentation in terms of generalization and sample complexity? We develop a general framework for investigating this question using Fourier analysis and the representation theory of finite groups. We show that, for a broad class of classical learning problems, 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. Our results provide a theoretical explanation for why partial augmentation can retain the statistical benefits of full augmentation despite enforcing symmetry only approximately, and shed light on a recently raised question in learning with symmetries: whether statistically optimal learning under general group invariances can be achieved using computationally scalable methods. Moreover, we prove a complementary impossibility result: enforcing exact invariance via data augmentation requires averaging over the entire group, and cannot be achieved by any strict subset when the hypothesis space is sufficiently expressive. Together, these results provide a unified perspective on full and partial data augmentation, as well as exact and approximate symmetry enforcement.