# Breaking Data Symmetry is Needed For Generalization in Feature Learning Kernels

arXiv: https://arxiv.org/abs/2604.00316 · HTML: https://mlsanity.com/paper/2604.00316/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 24 references that are in the map.
- First posted on arXiv 31 Mar 2026.

## Builds on (most cited first, 19 of 19)

- [Progress measures for grokking via mechanistic interpretability](https://mlsanity.com/paper/2301.05217/) (2023)
- [Grokking: Generalization Beyond Overfitting on Small Algorithmic Datasets](https://mlsanity.com/paper/2201.02177/) (2022)
- [Towards Understanding Grokking: An Effective Theory of Representation Learning](https://mlsanity.com/paper/2205.10343/) (2022)
- [Omnigrok: Grokking Beyond Algorithmic Data](https://mlsanity.com/paper/2210.01117/) (2022)
- [Explaining grokking through circuit efficiency](https://mlsanity.com/paper/2309.02390/) (2023)
- [Grokking as the Transition from Lazy to Rich Training Dynamics](https://mlsanity.com/paper/2310.06110/) (2023)
- [Grokking modular arithmetic](https://mlsanity.com/paper/2301.02679/) (2023)
- [Dichotomy of Early and Late Phase Implicit Biases Can Provably Induce Grokking](https://mlsanity.com/paper/2311.18817/) (2023)
- [Why Do You Grok? A Theoretical Analysis of Grokking Modular Addition](https://mlsanity.com/paper/2407.12332/) (2024)
- [Feature emergence via margin maximization: case studies in algebraic tasks](https://mlsanity.com/paper/2311.07568/) (2023)
- [Catapults in SGD: spikes in the training loss and their impact on generalization through feature learning](https://mlsanity.com/paper/2306.04815/) (2023)
- [Emergence in non-neural models: grokking modular arithmetic via average gradient outer product](https://mlsanity.com/paper/2407.20199/) (2024)
- [Grokking Group Multiplication with Cosets](https://mlsanity.com/paper/2312.06581/) (2023)
- [Average gradient outer product as a mechanism for deep neural collapse](https://mlsanity.com/paper/2402.13728/) (2024)
- [Mechanism of feature learning in convolutional neural networks](https://mlsanity.com/paper/2309.00570/) (2023)
- [Linear Recursive Feature Machines provably recover low-rank matrices](https://mlsanity.com/paper/2401.04553/) (2024)
- [Feature learning as alignment: a structural property of gradient descent in non-linear neural networks](https://mlsanity.com/paper/2402.05271/) (2024)
- [On the Ability of Deep Networks to Learn Symmetries from Data: A Neural Kernel Theory](https://mlsanity.com/paper/2412.11521/) (2024)
- [Learning words in groups: fusion algebras, tensor ranks and grokking](https://mlsanity.com/paper/2509.06931/) (2025)

## Built on by (most cited first, 1 of 1)

- [Spectral Entropy Collapse as a Phase Transition in Delayed Generalisation: An Interventional and Predictive Framework for Grokkin](https://mlsanity.com/paper/2604.13123/) (2026)

## Abstract (arXiv)

Grokking occurs when a model achieves high training accuracy but generalization to unseen test points happens long after that. This phenomenon was initially observed on a class of algebraic problems, such as learning modular arithmetic (Power et al., 2022). We study grokking on algebraic tasks in a class of feature learning kernels via the Recursive Feature Machine (RFM) algorithm (Radhakrishnan et al., 2024), which iteratively updates feature matrices through the Average Gradient Outer Product (AGOP) of an estimator in order to learn task-relevant features. Our main experimental finding is that generalization occurs only when a certain symmetry in the training set is broken. Furthermore, we empirically show that RFM generalizes by recovering the underlying invariance group action inherent in the data. We find that the learned feature matrices encode specific elements of the invariance group, explaining the dependence of generalization on symmetry.