On the Origins of Linear Representations in Large Language Models
Yibo Jiang Goutham Rajendran Pradeep Ravikumar Bryon Aragam Victor Veitch
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.