Linear Representation of Concepts in Large Language Models: A Geometric Perspective

This is a Plain English Papers summary of a research paper called Linear Representation of Concepts in Large Language Models: A Geometric Perspective. If you like these kinds of analysis, you should join AImodels.fyi or follow me on Twitter.

Overview

• The paper explores the "linear representation hypothesis" - the idea that high-level concepts are represented as linear directions in a representation space.
• It addresses two key questions: what does linear representation actually mean, and how do we make sense of geometric notions (like cosine similarity) in the representation space?
• The paper provides formal definitions of linear representation in the output (word) space and the input (sentence) space, and shows how these connect to linear probing and model steering, respectively.
• It introduces a particular (non-Euclidean) inner product that respects language structure, and uses this to unify different notions of linear representation.
• Experiments with the LLaMA-2 language model demonstrate the existence of linear representations of concepts, and the connections to interpretation and control.

Plain English Explanation

The paper explores the idea that high-level concepts in language models are represented as linear directions in a multi-dimensional space. This is known as the "linear representation hypothesis." The researchers wanted to better understand what this actually means and how it relates to the geometric properties of the representation space.

To do this, they provided formal definitions of linear representation in both the output (word) space and the input (sentence) space. These definitions show how linear representation connects to two important techniques: linear probing and model steering. Linear probing is a way to interpret what a model has learned, while model steering is a way to control a model's behavior.

The researchers also introduced a special type of inner product (a way of measuring similarity) that better captures the structure of language. Using this, they were able to unify different notions of linear representation and show how to construct useful probes and steering vectors.

Experiments on the LLaMA-2 language model provided evidence for the existence of linear representations of concepts, and demonstrated the connections to interpretation and control of the model's behavior.

Technical Explanation

The paper formalizes the "linear representation hypothesis" using the language of counterfactuals. It provides two definitions - one for the output (word) representation space, and one for the input (sentence) space.

The output space definition states that a concept is linearly represented if there exists a vector v such that the cosine similarity between v and the representation of any word w is monotonically related to the probability of w given the concept. This connects to linear probing, a technique for interpreting what a model has learned.

The input space definition states that a concept is linearly represented if there exists a vector v such that the projection of the sentence representation onto v is monotonically related to the probability of the sentence given the concept. This connects to model steering, a technique for controlling a model's behavior.

To make sense of geometric notions like cosine similarity in the representation space, the paper introduces a particular (non-Euclidean) inner product that respects language structure. This "causal inner product" is defined using counterfactual pairs, and allows the unification of different notions of linear representation.

Experiments on the LLaMA-2 language model demonstrate the existence of linear representations of concepts, and show how the formalization connects to interpretation and control techniques like those described in Representations as Language and Vectoring Languages.

Critical Analysis

The paper provides a rigorous formal framework for understanding linear representation in language models, and demonstrates its connections to important techniques like linear probing and model steering. However, the formalism relies on counterfactual pairs, which can be challenging to obtain in practice.

Additionally, the paper focuses on linear representations, but language models may also exhibit more complex, non-linear representations of concepts. Further research is needed to understand the full range of representational strategies used by large language models.

The experiments on LLaMA-2 provide evidence for the existence of linear representations, but it would be valuable to see this validated on a broader range of language models and tasks. The generalization of these findings to more diverse domains remains an open question.

Overall, the paper makes an important contribution to our understanding of language model representations, but there are still many open questions and avenues for further research in this area.

Conclusion

This paper introduces a formal framework for understanding the "linear representation hypothesis" in language models, which posits that high-level concepts are represented as linear directions in a multi-dimensional space. By defining linear representation in both the output (word) and input (sentence) spaces, the researchers were able to connect this idea to powerful techniques like linear probing and model steering.

The use of a novel "causal inner product" allowed the researchers to unify different notions of linear representation, and experiments on the LLaMA-2 model provided evidence for the existence of such linear representations. This work advances our fundamental understanding of how language models encode and utilize conceptual knowledge, with potential implications for model interpretation, control, and further advancements in the field.

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