On the Origins of Linear Representations in Large Language Models

TL;DR

This paper investigates why large language models develop linear representations of concepts, proposing a formal model and providing empirical evidence that the training objective and optimization bias promote such linearity.

Contribution

It introduces a latent variable model to explain the emergence of linear concept representations and validates the theory with experiments on LLaMA-2.

Findings

Linear representations emerge under data matching the model.

The training objective and gradient bias promote linearity.

Empirical validation on LLaMA-2 supports the theory.

Abstract

Recent works have argued that high-level semantic concepts are encoded "linearly" in the representation space of large language models. In this work, we study the origins of such linear representations. To that end, we introduce a simple latent variable model to abstract and formalize the concept dynamics of the next token prediction. We use this formalism to show that the next token prediction objective (softmax with cross-entropy) and the implicit bias of gradient descent together promote the linear representation of concepts. Experiments show that linear representations emerge when learning from data matching the latent variable model, confirming that this simple structure already suffices to yield linear representations. We additionally confirm some predictions of the theory using the LLaMA-2 large language model, giving evidence that the simplified model yields generalizable insights.