# Grokking Finite-Dimensional Algebra

arXiv: https://arxiv.org/abs/2602.19533 · HTML: https://mlsanity.com/paper/2602.19533/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 28 references that are in the map.
- First posted on arXiv 23 Feb 2026.

## Builds on (most cited first, 20 of 20)

- [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)
- [Implicit Regularization in Deep Matrix Factorization](https://mlsanity.com/paper/1905.13655/) (2019)
- [Implicit Regularization in Matrix Factorization](https://mlsanity.com/paper/1705.09280/) (2017)
- [On the Optimization of Deep Networks: Implicit Acceleration by Overparameterization](https://mlsanity.com/paper/1802.06509/) (2018)
- [Towards Understanding Grokking: An Effective Theory of Representation Learning](https://mlsanity.com/paper/2205.10343/) (2022)
- [Towards Resolving the Implicit Bias of Gradient Descent for Matrix Factorization: Greedy Low-Rank Learning](https://mlsanity.com/paper/2012.09839/) (2020)
- [A Toy Model of Universality: Reverse Engineering How Networks Learn Group Operations](https://mlsanity.com/paper/2302.03025/) (2023)
- [Implicit Regularization of Discrete Gradient Dynamics in Linear Neural Networks](https://mlsanity.com/paper/1904.13262/) (2019)
- [Implicit Regularization in Deep Learning May Not Be Explainable by Norms](https://mlsanity.com/paper/2005.06398/) (2020)
- [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)
- [The Implicit Bias of Depth: How Incremental Learning Drives Generalization](https://mlsanity.com/paper/1909.12051/) (2019)
- [Predicting Grokking Long Before it Happens: A look into the loss landscape of models which grok](https://mlsanity.com/paper/2306.13253/) (2023)
- [Grokking in Linear Estimators -- A Solvable Model that Groks without Understanding](https://mlsanity.com/paper/2310.16441/) (2023)
- [Grokking Group Multiplication with Cosets](https://mlsanity.com/paper/2312.06581/) (2023)
- [Grokking Beyond the Euclidean Norm of Model Parameters](https://mlsanity.com/paper/2506.05718/) (2025)
- [Grokking as Compression: A Nonlinear Complexity Perspective](https://mlsanity.com/paper/2310.05918/) (2023)
- [Semantic Composition and Decomposition: From Recognition to Generation](https://mlsanity.com/paper/1405.7908/) (2014)

## Built on by (most cited first, 2 of 2)

- [From Symmetry to Invariance: Learning Galois Equivalent Representations in Finite Fields](https://mlsanity.com/paper/2608.22513/) (2026)
- [A Stochastic--Geometric Theory of Scaling Laws in Grokking](https://mlsanity.com/paper/2606.30388/) (2026)

## Abstract (arXiv)

This paper investigates the grokking phenomenon, which refers to the sudden transition from a long memorization to generalization observed during neural networks training, in the context of learning multiplication in finite-dimensional algebras (FDA). While prior work on grokking has focused mainly on group operations, we extend the analysis to more general algebraic structures, including non-associative, non-commutative, and non-unital algebras. We show that learning group operations is a special case of learning FDA, and that learning multiplication in FDA amounts to learning a bilinear product specified by the algebra's structure tensor. For algebras over the reals, we connect the learning problem to matrix factorization with an implicit low-rank bias, and for algebras over finite fields, we show that grokking emerges naturally as models must learn discrete representations of algebraic elements. This leads us to experimentally investigate the following core questions: (i) how do algebraic properties such as commutativity, associativity, and unitality influence both the emergence and timing of grokking, (ii) how structural properties of the structure tensor of the FDA, such as sparsity and rank, influence generalization, and (iii) to what extent generalization correlates with the model learning latent embeddings aligned with the algebra's representation. Our work provides a unified framework for grokking across algebraic structures and new insights into how mathematical structure governs neural network generalization dynamics.