Latent Semantic Manifolds in Large Language Models

TL;DR

This paper introduces a geometric framework for understanding how large language models encode semantics in continuous spaces, revealing fundamental limits and properties of their internal representations.

Contribution

It develops a mathematical model interpreting LLM hidden states as points on a Riemannian manifold, and proves key theorems about semantic distortion and expressibility gaps.

Findings

Universal hourglass intrinsic dimension profiles across models

Linear scaling law for the semantic expressibility gap

Persistent boundary-proximal representations invariant to scale

Abstract

Large Language Models (LLMs) perform internal computations in continuous vector spaces yet produce discrete tokens -- a fundamental mismatch whose geometric consequences remain poorly understood. We develop a mathematical framework that interprets LLM hidden states as points on a latent semantic manifold: a Riemannian submanifold equipped with the Fisher information metric, where tokens correspond to Voronoi regions partitioning the manifold. We define the expressibility gap, a geometric measure of the semantic distortion from vocabulary discretization, and prove two theorems: a rate-distortion lower bound on distortion for any finite vocabulary, and a linear volume scaling law for the expressibility gap via the coarea formula. We validate these predictions across six transformer architectures (124M-1.5B parameters), confirming universal hourglass intrinsic dimension profiles, smooth curvature structure, and linear gap scaling with slopes 0.87-1.12 (R^2 > 0.985). The margin distribution across models reveals a persistent hard core of boundary-proximal representations invariant to scale, providing a geometric decomposition of perplexity. We discuss implications for architecture design, model compression, decoding strategies, and scaling laws