The Linear Representation Hypothesis and the Geometry of Large Language Models
This work addresses foundational questions in understanding and manipulating large language models, offering theoretical insights and practical tools for interpretability and control, though it is incremental in formalizing existing informal ideas.
The paper formalizes the 'linear representation hypothesis' by defining two notions of linear representation using counterfactuals and connecting them to linear probing and model steering, and introduces a causal inner product to unify geometric notions in representation spaces, with experiments on LLaMA-2 demonstrating linear representations and their role in interpretation and control.
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.