Dynamic Manifold Evolution Theory: Modeling and Stability Analysis of Latent Representations in Large Language Models

2025-05-24

AI Summary

This work addresses the quantitative relationship between hidden-state dynamics and generated text quality during large language model (LLM) inference. We formulate text generation as a controlled dynamical system on a semantic manifold, approximating hidden-state updates via continuous-time dynamics to uncover how attention and residual connections jointly drive semantic evolution. We introduce the first theoretical framework for hidden-space dynamic manifolds, defining three quantifiable metrics—state continuity, clustering quality, and topological persistence—based on Lyapunov stability theory. These metrics establish a causal explanatory chain linking trajectory properties in the hidden space to textual fluency, grammaticality, and semantic coherence. Empirical evaluation confirms the theoretical predictions’ accuracy across diverse LLMs and tasks. Moreover, our framework yields interpretable, reproducible principles for balancing creativity and consistency in decoding strategies, grounded in geometric and dynamical properties of the latent space.

Technology Category

• Natural Language Processing: (Large) Language Models
• Machine Learning: Large Multimodal Models (LMMs)
• Cognitive Modeling & Cognitive Systems: Computational Creativity

Application Category

• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• User Modeling, Personalization and Recommendation: Large Language Models (LLM) for user modeling and recommendation
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems

Abstract

We introduce Dynamic Manifold Evolution Theory (DMET),a unified framework that models large language model generation as a controlled dynamical system evolving on a low_dimensional semantic manifold. By casting latent_state updates as discrete time Euler approximations of continuous dynamics, we map intrinsic energy_driven flows and context_dependent forces onto Transformer components (residual connections, attention, feed-forward networks). Leveraging Lyapunov stability theory We define three empirical metrics (state continuity, clustering quality, topological persistence) that quantitatively link latent_trajectory properties to text fluency, grammaticality, and semantic coherence. Extensive experiments across decoding parameters validate DMET's predictions and yield principled guidelines for balancing creativity and consistency in text generation.

Problem

• Modeling LLM generation as controlled dynamical system
• Linking latent-trajectory properties to text quality
• Balancing creativity and consistency in text generation

Innovation

• Modeling LLM generation as controlled dynamical system
• Mapping energy-driven flows to Transformer components
• Linking latent-trajectory properties to text quality