The paper shows that the geometric patterns seen in language model embeddings—such as circles for months and saddle-shaped manifolds—arise from group-invariant statistics in word co‑occurrence data. By extending previous work on translation symmetry to arbitrary finite, compact, and homogeneous groups, the authors prove that embeddings correspond to matrix elements of the irreducible representations of the symmetry group. They validate this theory experimentally with the cyclic group <Z_{12}> for months, a dihedral group for musical chords, and a spherical‑harmonic embedding for celestial objects.
arXiv:2602. 15029v3 Announce Type: replace Abstract: The internal representations learned by language models consistently exhibit striking geometric structure: calendar months organize into a circle, historical years form a smooth one-dimensional manifold, and cities' latitudes and longitudes can be decoded using a linear probe.
The paper investigates how large language models encode and use relational information among tokens across transformer layers. By analyzing activations from prompts that require inferring relationships among three cyclic tokens (months, hours, weekdays, musical notes), the authors find a consistent layerwise progression: intermediate layers capture pairwise relationships, while later layers encode the full three‑token relationship to predict the next token. They also identify geometrically structured token relationships that do not influence prediction, and show that constraining models to use only causally relevant joint representations improves next‑token accuracy.
arXiv:2607. 07047v1 Announce Type: cross Abstract: Understanding the geometric structure of pre-trained language model embeddings matters for interpretability and safety.
arXiv:2608.31045v1 Announce Type: new Abstract: Rotational symmetry is one of the most important structural principles in machine learning on 3D data. In applications ranging from physics and materia...
arXiv:2606. 24418v1 Announce Type: new Abstract: Data augmentation is a simple and model-agnostic approach for exploiting known invariances in learning problems.
arXiv:2606. 14612v1 Announce Type: cross Abstract: We show that the three movements of Beethoven's "Moonlight Sonata" (Op.
The Platonic Representation Hypothesis (PRH) claims that independently trained models converge on a shared statistical model of reality, yet recent work finds only weak pointwise similarity between mo...
arXiv:2602. 24264v2 Announce Type: replace-cross Abstract: Compositional generalization, the ability to recognize familiar parts in novel contexts, is a defining property of intelligent systems.
arXiv:2609.24209v1 Announce Type: new Abstract: The Platonic Representation Hypothesis (PRH) claims that independently trained models converge on a shared statistical model of reality, yet recent wor...
The paper shows that language models trained on natural text develop number representations that exhibit periodic features with dominant periods at T = 2, 5, 10. It identifies a two‑tiered hierarchy: all models learn Fourier‑domain spikes at these periods, but only some acquire geometrically separable features that allow linear classification of numbers modulo T. The study demonstrates that data, architecture, optimizer, and tokenizer influence whether these separable features emerge, and that models can learn them either from co‑occurrence signals in language or from multi‑token addition tasks, illustrating convergent evolution across diverse models.
arXiv:2609.15083v1 Announce Type: new Abstract: Mixed-curvature representation learning seeks to capture rich geometric structures that cannot be adequately modeled by a single curvature regime. Exis...
arXiv:2602. 14419v2 Announce Type: cross Abstract: This paper reformulates Transformer/Attention mechanisms in Large Language Models (LLMs) through measure theory and frequency analysis, theoretically demonstrating that hallucination is an inevitable structural limitation.