The Origins of Representation Manifolds in Large Language Models

TL;DR

This paper explores how features are represented as manifolds in large language models, linking geometric properties in embedding space to conceptual relatedness, and validates the theory on text embeddings.

Contribution

It introduces a manifold-based model of feature representation in neural embeddings, extending beyond linear assumptions to capture continuous, multidimensional features.

Findings

Cosine similarity encodes feature geometry on manifolds

Manifold paths relate to concept relatedness

Theory validated on LLM text embeddings

Abstract

There is a large ongoing scientific effort in mechanistic interpretability to map embeddings and internal representations of AI systems into human-understandable concepts. A key element of this effort is the linear representation hypothesis, which posits that neural representations are sparse linear combinations of `almost-orthogonal' direction vectors, reflecting the presence or absence of different features. This model underpins the use of sparse autoencoders to recover features from representations. Moving towards a fuller model of features, in which neural representations could encode not just the presence but also a potentially continuous and multidimensional value for a feature, has been a subject of intense recent discourse. We describe why and how a feature might be represented as a manifold, demonstrating in particular that cosine similarity in representation space may encode the intrinsic geometry of a feature through shortest, on-manifold paths, potentially answering the question of how distance in representation space and relatedness in concept space could be connected. The critical assumptions and predictions of the theory are validated on text embeddings and token activations of large language models.