Kiho Park

Kiho Park, Yo Joong Choe, Victor Veitch

Informally, the 'linear representation hypothesis' is the idea that high-level concepts are represented linearly as directions in some representation space. In this paper, we address two closely related questions: What does "linear representation" actually mean? And, how do we make sense of geometric notions (e.g., cosine similarity or projection) in the representation space? To answer these, we use the language of counterfactuals to give two formalizations of "linear representation", one in the output (word) representation space, and one in the input (sentence) space. We then prove these connect to linear probing and model steering, respectively. To make sense of geometric notions, we use the formalization to identify a particular (non-Euclidean) inner product that respects language structure in a sense we make precise. Using this causal inner product, we show how to unify all notions of linear representation. In particular, this allows the construction of probes and steering vectors using counterfactual pairs. Experiments with LLaMA-2 demonstrate the existence of linear representations of concepts, the connection to interpretation and control, and the fundamental role of the choice of inner product.

Kiho Park, Todd Nief, Yo Joong Choe et al.

This paper concerns the question of how AI systems encode semantic structure into the geometric structure of their representation spaces. The motivating observation of this paper is that the natural geometry of these representation spaces should reflect the way models use representations to produce behavior. We focus on the important special case of representations that define softmax distributions. In this case, we argue that the natural geometry is information geometry. Our focus is on the role of information geometry on semantic encoding and the linear representation hypothesis. As an illustrative application, we develop "dual steering", a method for robustly steering representations to exhibit a particular concept using linear probes. We prove that dual steering optimally modifies the target concept while minimizing changes to off-target concepts. Empirically, we find that dual steering enhances the controllability and stability of concept manipulation.

Mark Muchane, Sean Richardson, Kiho Park et al.

Sparse dictionary learning (and, in particular, sparse autoencoders) attempts to learn a set of human-understandable concepts that can explain variation on an abstract space. A basic limitation of this approach is that it neither exploits nor represents the semantic relationships between the learned concepts. In this paper, we introduce a modified SAE architecture that explicitly models a semantic hierarchy of concepts. Application of this architecture to the internal representations of large language models shows both that semantic hierarchy can be learned, and that doing so improves both reconstruction and interpretability. Additionally, the architecture leads to significant improvements in computational efficiency.

Kiho Park, Yo Joong Choe, Yibo Jiang et al.

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.