Latent Semantic Manifolds in Large Language Models
arXiv:2603.22301v1 Announce Type: new 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 slope
arXiv:2603.22301v1 Announce Type: new 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
Executive Summary
This article presents a novel mathematical framework for understanding the internal computations of Large Language Models (LLMs) by interpreting hidden states as points on a latent semantic manifold. The authors develop a geometric measure of the semantic distortion from vocabulary discretization, known as the expressibility gap, and prove two theorems that provide a rate-distortion lower bound and a linear volume scaling law. The framework is validated across six transformer architectures, revealing universal hourglass intrinsic dimension profiles, smooth curvature structure, and linear gap scaling. The results have significant implications for LLM design, compression, and decoding strategies.
Key Points
- ▸ Development of a mathematical framework for understanding LLM internal computations
- ▸ Introduction of the latent semantic manifold and the expressibility gap
- ▸ Proof of two theorems providing a rate-distortion lower bound and a linear volume scaling law
Merits
Strength
The framework provides a novel and comprehensive understanding of LLM internal computations, enabling the development of more effective LLM design, compression, and decoding strategies.
Demerits
Limitation
The framework relies on the assumption of a finite vocabulary, which may not be applicable in real-world scenarios where the vocabulary is vast and dynamic.
Limitation
The validation of the framework is limited to six transformer architectures, and it is unclear whether the results are generalizable to other architectures or models.
Expert Commentary
This article marks a significant advancement in the field of large language models, providing a novel mathematical framework for understanding their internal computations. The development of the latent semantic manifold and the expressibility gap offers a powerful tool for analyzing and improving LLM performance. However, further research is needed to explore the limitations and generalizability of the framework. Additionally, the framework's implications for model interpretability, explainability, and efficient training and inference methods are significant and warrant further investigation.
Recommendations
- ✓ Future research should focus on extending the framework to other architectures and models, as well as exploring its applicability to real-world scenarios with vast and dynamic vocabularies.
- ✓ Developers should consider incorporating the framework into LLM design, compression, and decoding strategies to improve performance and efficiency in natural language processing tasks.
Sources
Original: arXiv - cs.LG
