How Data Augmentation Shapes Neural Representations

Grokking is a phenomenon in which neural networks initially memorize training data and only later exhibit strong generalization after prolonged optimization. Despite extensive recent study, the factors influencing the emergence and timing of grokking remain incompletely understood. We investigate the relationship between representation geometry and delayed generalization. We find that dimensionality collapse consistently precedes the onset of grokking in all evaluated settings. Motivated by these observations, we introduce Geometric Dimensionality Regularization (GeomDR), a simple spectral regularizer that modifies the effective dimensionality of hidden representations during training. Across modular addition, modular division, and permutation composition tasks, GeomDR consistently alters grokking dynamics and can substantially accelerate the onset of generalization depending on the intervention schedule and target dimensionality. In several settings, grokking is accelerated by up to 52 times relative to standard AdamW training. Similar qualitative effects are observed in both multilayer perceptrons and transformers. Together, these results suggest that representation geometry can serve as an effective control signal for grokking and provide evidence that geometric interventions offer a practical approach for studying and influencing delayed generalization in neural networks.

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.

Diffusion geometry is a manifold learning framework that uses random walks defined by Markov transition matrices to characterize the geometry of a dataset at multiple scales. We use diffusion geometry for neural representations, incorporating tools from multi-view learning into this field for the first time. Our key technical observation is that a broad class of similarity measures based on representational similarity matrices (RSMs) admits a closed-form equivalent formulation in terms of row-stochastic Markov matrices, opening the door to manipulations from diffusion geometry. As a first application, we develop multi-scale variants of Centered Kernel Alignment and Distance Correlation, which utilise the $t^{th}$ power of the underlying transition matrix to probe the data geometry at adjustable diffusion scales. Going further, we introduce variants of these measures which fuse the Markov matrices of several layers via alternating diffusion into a single operator that captures the network's joint sample geometry, allowing similarity to be computed across multiple layers and shifting the comparison from layer-to-layer to network-to-network. We perform extensive numerical experiments, evaluating our measures on the Representational Similarity (ReSi) benchmark comprising 14 architectures trained on 7 datasets across three different domains. Our methods achieve SoTA results in accuracy and output correlation for both language and vision tasks across different models. We furthermore show SoTA performance on an additional benchmark evaluating on out-of-distribution data.