# Learning words in groups: fusion algebras, tensor ranks and grokking

arXiv: https://arxiv.org/abs/2509.06931 · HTML: https://mlsanity.com/paper/2509.06931/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 33 references that are in the map.
- First posted on arXiv 8 Sep 2025.

## Builds on (most cited first, 18 of 18)

- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [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)
- [Hidden Progress in Deep Learning: SGD Learns Parities Near the Computational Limit](https://mlsanity.com/paper/2207.08799/) (2022)
- [Omnigrok: Grokking Beyond Algorithmic Data](https://mlsanity.com/paper/2210.01117/) (2022)
- [A Toy Model of Universality: Reverse Engineering How Networks Learn Group Operations](https://mlsanity.com/paper/2302.03025/) (2023)
- [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)
- [The Slingshot Mechanism: An Empirical Study of Adaptive Optimizers and the Grokking Phenomenon](https://mlsanity.com/paper/2206.04817/) (2022)
- [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)
- [Deep Networks Always Grok and Here is Why](https://mlsanity.com/paper/2402.15555/) (2024)
- [Grokking as a First Order Phase Transition in Two Layer Networks](https://mlsanity.com/paper/2310.03789/) (2023)
- [Why Do You Grok? A Theoretical Analysis of Grokking Modular Addition](https://mlsanity.com/paper/2407.12332/) (2024)
- [To grok or not to grok: Disentangling generalization and memorization on corrupted algorithmic datasets](https://mlsanity.com/paper/2310.13061/) (2023)
- [Grokking Modular Polynomials](https://mlsanity.com/paper/2406.03495/) (2024)
- [Deep Grokking: Would Deep Neural Networks Generalize Better?](https://mlsanity.com/paper/2405.19454/) (2024)

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

- [Provable Scaling Laws of Feature Emergence from Learning Dynamics of Grokking](https://mlsanity.com/paper/2509.21519/) (2025)
- [Breaking Data Symmetry is Needed For Generalization in Feature Learning Kernels](https://mlsanity.com/paper/2604.00316/) (2026)
- [The Rank the Task Demands: A Causal Rank Law for Matrix Memories Trained on Group Composition](https://mlsanity.com/paper/2609.12259/) (2026)
- [Tracking States or Tracking Cosets? An Algebraic Account of Learned State Tracking](https://mlsanity.com/paper/2609.29951/) (2026)

## Abstract (arXiv)

In this work, we demonstrate that a simple two-layer neural network with standard activation functions can learn an arbitrary word operation in any finite group, provided sufficient width is available and exhibits grokking while doing so. To explain the mechanism by which this is achieved, we reframe the problem as that of learning a particular 3-tensor, which we show is typically of low rank. A key insight is that low-rank implementations of this tensor can be obtained by decomposing it along triplets of basic self-conjugate representations of the group and leveraging the fusion structure to rule out many components. Focusing on a phenomenologically similar but more tractable surrogate model, we show that the network is able to find such low-rank implementations (or approximations thereof), thereby using limited width to approximate the word-tensor in a generalizable way. In the case of the simple multiplication word, we further elucidate the form of these low-rank implementations, showing that the network effectively implements efficient matrix multiplication in the sense of Strassen. Our work also sheds light on the mechanism by which a network reaches such a solution under gradient descent.