The Geometry of Categorical and Hierarchical Concepts in Large Language Models
Abstract
The linear representation hypothesis is the informal idea that semantic concepts are encoded as linear directions in the representation spaces of large language models (LLMs). Previous work has shown how to make this notion precise for representing binary concepts that have natural contrasts (e.g., male, female) as _directions_ in representation space. However, many natural concepts do not have natural contrasts (e.g., whether the output is about an animal). In this work, we show how to extend the formalization of the linear representation hypothesis to represent features (e.g., is_animal) as _vectors_. This allows us to immediately formalize the representation of categorical concepts as polytopes in the representation space. Further, we use the formalization to prove a relationship between the hierarchical structure of concepts and the geometry of their representations. We validate these theoretical results on the Gemma and LLaMA-3 large language models, estimating representations for 900+ hierarchically related concepts using data from WordNet.
Cite
Text
Park et al. "The Geometry of Categorical and Hierarchical Concepts in Large Language Models." International Conference on Learning Representations, 2025.Markdown
[Park et al. "The Geometry of Categorical and Hierarchical Concepts in Large Language Models." International Conference on Learning Representations, 2025.](https://mlanthology.org/iclr/2025/park2025iclr-geometry/)BibTeX
@inproceedings{park2025iclr-geometry,
title = {{The Geometry of Categorical and Hierarchical Concepts in Large Language Models}},
author = {Park, Kiho and Choe, Yo Joong and Jiang, Yibo and Veitch, Victor},
booktitle = {International Conference on Learning Representations},
year = {2025},
url = {https://mlanthology.org/iclr/2025/park2025iclr-geometry/}
}