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arXiv · 2605.06352

Topological Signatures of Grokking

Yifan Tang·Qiquan Wang·Inés García-Redondo·Anthea Monod
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Abstract

We study the grokking phenomenon through the lens of topology. Using persistent homology on point clouds derived from the embedding matrices of a range of models trained on modular arithmetic with varying primes, we identify a clear and consistent topological signature of grokking: a sharp increase in both the maximum and total persistence of first homology ($H_1$). Persistence diagrams reveal the emergence of a dominant long-lived topological feature together with increasingly structured secondary features, reflecting the underlying cyclic structure of the task. Compared to existing spectral and geometric diagnostics -- specifically, Fourier analysis and local intrinsic dimension -- persistent homology provides a unified geometric and topological characterization of representation learning, capturing both local and global multi-scale structure. Ablations across data regimes and control settings show that these topological transitions are tied to generalization rather than memorization. Our results suggest that persistent homology offers a principled and interpretable framework for analyzing how neural networks internalize latent structure during training.

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Yifan Tang, Qiquan Wang, Inés García-Redondo, Anthea Monod. 2026-05-07. Topological Signatures of Grokking. https://arxiv.org/abs/2605.06352

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