Figure: Illustration of Unified Concept Representations Across Embedding and Unembedding Spaces via the Causal Inner Product

(Figure courtesy: Kiho Park)

Figure: Comparison of Causal vs. Euclidean Inner Products Between Counterfactually Defined Concepts in the Gemma-2B LLM

(Figure courtesy: Kiho Park)

Abstract

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 and 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 (context) space. We then prove that 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.

Citation

Park, K., Choe, Y. J., & Veitch, V. (2024). The Linear Representation Hypothesis and the Geometry of Large Language Models. Proceedings of the 41st International Conference on Machine Learning, PMLR 235:39643-39666.

@inproceedings{park2024linear,
  title={The Linear Representation Hypothesis and the Geometry of Large Language Models},
  author={Park, Kiho and Choe, Yo Joong and Veitch, Victor},
  booktitle={Proceedings of the 41st International Conference on Machine Learning},
  pages={39643--39666},
  year={2024},
  editor={Salakhutdinov, Ruslan and Kolter, Zico and Heller, Katherine and Weller, Adrian and Oliver, Nuria and Scarlett, Jonathan and Berkenkamp, Felix},
  volume={235},
  series={Proceedings of Machine Learning Research},
  publisher={PMLR},
}