Entropy, Thermodynamics and the Geometrization of the Language Model
Wenzhe Yang
TL;DR
The paper proposes a rigorous framework that unifies language modeling with concepts from set theory, analysis, information theory, thermodynamics, and differential geometry. It defines causal and predicative language models, introduces a moduli space of distributions, and formalizes an entropy-based view of information flow, including zero points and an AGI conjecture. A thermodynamic interpretation is developed via Boltzmann distribution, partition functions, and free energy, culminating in a geometrization of language models through Boltzmann manifolds and an epsilon-geometrization that maps complex distribution spaces to computable manifolds. The framework recasts current LLMs as special cases within this geometrization, linking transformer architectures to energy-based and geometric descriptions, and outlines open problems on optimal manifolds and cross-language representations with potential implications for AGI and model design.
Abstract
In this paper, we discuss how pure mathematics and theoretical physics can be applied to the study of language models. Using set theory and analysis, we formulate mathematically rigorous definitions of language models, and introduce the concept of the moduli space of distributions for a language model. We formulate a generalized distributional hypothesis using functional analysis and topology. We define the entropy function associated with a language model and show how it allows us to understand many interesting phenomena in languages. We argue that the zero points of the entropy function and the points where the entropy is close to 0 are the key obstacles for an LLM to approximate an intelligent language model, which explains why good LLMs need billions of parameters. Using the entropy function, we formulate a conjecture about AGI. Then, we show how thermodynamics gives us an immediate interpretation to language models. In particular we will define the concepts of partition function, internal energy and free energy for a language model, which offer insights into how language models work. Based on these results, we introduce a general concept of the geometrization of language models and define what is called the Boltzmann manifold. While the current LLMs are the special cases of the Boltzmann manifold.