The Geometry of Categorical and Hierarchical Concepts in Large Language Models

The Geometry of Categorical and Hierarchical Concepts in LLMs

In their study on the geometric representation of semantic meaning in LLMs, Park et al. investigate the fundamental questions concerning the representation of categorical and hierarchical concepts within these models. Understanding the internal representation of these high-level semantic concepts is crucial for the interpretability and control of LLMs.

Key Contributions

This paper addresses how categorical concepts, such as {mammal, bird, reptile, fish}, are represented and how hierarchical relations between these concepts, e.g., how 'dog' being a subset of 'mammal', are encoded. The study builds on the linear representation hypothesis that posits high-level concepts are linearly encoded in the representation spaces of LLMs.

Revamping Linear Representations:

• The authors extend the previous notion of linear representation of binary concepts as directions to the representation as vectors. This transformation allows for the composition of representations using vector operations.

Hierarchical Orthogonality:

• They demonstrate that the hierarchical relationships between concepts are encoded geometrically as orthogonality in the representation space. Specifically, they show that subordinate concepts reside in orthogonal subspaces relative to their superordinate category.

Representation of Categorical Concepts:

• The paper introduces a framework to represent categorical variables as polytopes, with representations of natural concepts forming a simplex. For example, the concept of 'animal' is represented by the convex hull of the vector representations of its subcategories (mammal, bird, reptile, fish).

Empirical Validation:

• The theoretical outcomes are empirically validated on the Gemma large language model by estimating representations for 957 hierarchically related concepts using data from WordNet.

Experimental Setup and Findings

Canonical Representation Space

To align the embedding and unembedding representations, the authors use a canonical transformation, which ensures the Euclidean inner product becomes a causal inner product. This transformation is crucial for employing vector operations with semantic fidelity in the unified space.

WordNet Hierarchy Analysis

Existence of Vector Representations:

• Using WordNet to define a comprehensive set of binary features, the authors verify the existence of vector representations. They employ Linear Discriminant Analysis (LDA) to estimate vector representations and confirm that projected test word vectors agree with theoretical properties.

Hierarchical Orthogonality:

• The cosine similarity between the vector representations of concepts reveals that WordNet's hierarchical structure is mirrored in the representation space. Parent-child and child-grandparent vectors are shown to be orthogonal, validating the theoretical predictions.

Implications and Future Directions

The findings of this research lay substantial theoretical groundwork for interpretability in AI. The results not only reinforce the existence of linear representations but also elucidate how hierarchical semantic structures are geometrically encoded in LLMs. These insights point to several future research avenues:

Hierarchical Semantic Interpretability:

• Current interpretability models should be revised to account for hierarchical semantics. Methods should respect the orthogonal direct sum structure of categorical variables to ensure clean separability and interpretability.

Internal Layer Geometry:

• While the study focuses on final layer representations, it opens up questions about the geometric structure within internal layers. Investigations into canonical transformations for internal layers could yield deeper insights.

Efficient Hyperbolic Representations:

• Given the observed simplex structure for categorical concepts, exploring hyperbolic geometries for more efficient LLM representations could be promising.

This comprehensive inquiry into the geometric representation of hierarchical and categorical concepts within LLMs provides foundational insights that may drive significant advancements in the design and interpretability of intelligent systems. By establishing a clear connection between vector space operations and semantic structures, Park et al. propel the broader research agenda in understanding and manipulating the internal representations of language models.