The Geometry of Categorical and Hierarchical Concepts in Large Language Models

TL;DR

This paper formalizes how large language models encode hierarchical and categorical concepts as geometric structures, extending previous linear hypothesis work to non-contrastive features and validating with empirical data.

Contribution

It introduces a formal framework for representing hierarchical and categorical concepts as geometric structures in LLMs, extending the linear representation hypothesis.

Findings

Hierarchical concepts correspond to specific geometric structures in LLM representations

The formalization links concept hierarchy with geometric relationships in embedding space

Validated on Gemma and LLaMA-3 models with 900+ concepts from WordNet

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.