Group-Invariant Statistics Determine Embedding Geometry: Harmonic Analysis of Representations from Bach to the Night Sky
The representations that language models learn for concepts such as months, weekdays, and places display consistent geometric structure: circles and saddle-shaped "Pringle" manifolds. Recent work traced these structures to $\textit{translation symmetry}$ in word co-occurrence statistics, deriving the observed Fourier geometry when co-occurrence depends only on distance on an abelian lattice of concepts. We demonstrate that more general notions of symmetry lead to equally structured predictions. Considering symmetries defined by arbitrary finite groups, compact groups, and homogeneous spaces, we prove that whenever the co-occurrence statistics of a word family are invariant under a group $...
| Authors | Liam Storan, Andreas Tolias, Nina Miolane |
|---|---|
| Primary category | cs.LG |
| Published | 30 September 2026 |
| Last updated at source | 30 September 2026 |
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