arXiv:2311.03658 · v2
The Linear Representation Hypothesis and the Geometry of Large Language Models
摘要
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.
作者备注：Accepted for a presentation at ICML 2024 and an oral presentation at NeurIPS 2023 Workshop on Causal Representation Learning. Code is available at https://github.com/KihoPark/linear_rep_geometry
期刊信息：In Proceedings of the 41st International Conference on Machine Learning (ICML), 2024