Why LLMs Form Geometric Representations
Free while signed in. Answers cite the passages they came from.
LLMs spontaneously form striking geometric structures in their internal representations: calendar months organize into circles, historical years form spirals, and spatial coordinates align to recoverable manifolds. This paper proves these patterns are not the product of deep learning dynamics but emerge directly from symmetries in natural language statistics.
Translation symmetry as the root cause: The frequency with which any two months co-occur in text depends only on the time interval between them, not the months themselves. The authors prove this translation symmetry in co-occurrence statistics is sufficient to force circular geometry in learned representations.
Analytical derivation of manifold geometry: Rather than just observing geometric structure post-hoc, the paper derives the exact manifold geometry from data statistics. For cyclic concepts like months or days of the week, the proof shows circular representations emerge as the optimal encoding under symmetric co-occurrence distributions.
Spirals and rippled manifolds for continuums: Representations of continuous concepts like historical years or number lines organize into compact 1D manifolds with characteristic extrinsic curvature. These “rippled” structures are analytically predicted by the framework when the underlying latent variable is non-cyclic.
Universal origin: The robustness of these geometric representations across different model architectures suggests a universal mechanism. Representational manifolds emerge whenever co-occurrence statistics are controlled by an underlying latent variable, regardless of model size or training details.
Get next week’s papers.
The same picks and the same summaries, in your inbox. Free, and 176 issues deep.
Subscribe on Substack