Empirical Investigation of Latent Representational Dynamics in Large Language Models: A Manifold Evolution Perspective

Abstract: This paper introduces the Dynamical Manifold Evolution Theory (DMET), a conceptual framework that models LLM generation as a continuous trajectory evolving on a low-dimensional semantic manifold. The theory characterizes latent dynamics through three interpretable metrics-state continuity ($C$), attractor compactness ($Q$), and topological persistence ($P$)-which jointly capture the smoothness, stability, and structure of representation evolution. Empirical analyses across multiple Transformer architectures reveal consistent links between these latent dynamics and text quality: smoother trajectories correspond to greater fluency, and richer topological organization correlates with enhanced coherence. Different models exhibit distinct dynamical regimes, reflecting diverse strategies of semantic organization in latent space. Moreover, decoding parameters such as temperature and top-$p$ shape these trajectories in predictable ways, defining a balanced region that harmonizes fluency and creativity. As a phenomenological rather than first-principles framework, DMET provides a unified and testable perspective for interpreting, monitoring, and guiding LLM behavior, offering new insights into the interplay between internal representation dynamics and external text generation quality.

• The paper introduces the DMET framework that models LLM generation as trajectories on a semantic manifold with metrics for continuity, attractor compactness, and topological persistence.
• It empirically links decoding parameters like temperature and top-p to measurable performance metrics such as perplexity, fluency, and grammatical consistency.
• The study offers actionable strategies for balancing fluency and creativity while highlighting challenges and opportunities for real-time latent dynamics analysis.

Empirical Investigation of Latent Representational Dynamics in LLMs: A Manifold Evolution Perspective

Introduction

The paper "Empirical Investigation of Latent Representational Dynamics in LLMs: A Manifold Evolution Perspective" introduces the Dynamical Manifold Evolution Theory (DMET) as a novel framework to conceptualize the generative process of LLMs. It models LLM generation as a trajectory evolving on a low-dimensional semantic manifold, characterized by three metrics: state continuity ($C$), attractor compactness ($Q$), and topological persistence ($P$). This representation allows for a comprehensive understanding of how latent dynamics relate to text quality, including fluency, coherence, and grammaticality. The framework extends previous work by providing a unified, continuous view of LLM dynamics and suggesting that decoding parameters like temperature and top-$p$ can predictably influence these trajectories.

Dynamical Manifold Evolution Theory (DMET) Framework

DMET models LLM generation as a controlled dynamical system on a semantic manifold. The central assumptions are:

1. Manifold Hypothesis: Latent representations exist on a low-dimensional manifold within a high-dimensional space.
2. Continuity: Text generation corresponds to a smooth trajectory rather than discrete jumps.
3. Attractors: The manifold is organized into attractor basins that correspond to coherent semantic states.

   Figure 1: Overview of the DMET framework where trajectories evolve on a low-dimensional semantic manifold under intrinsic energy gradients and context-driven forces.

The mathematical formulation uses a continuous-time model and maps the manifold dynamics onto Transformer architecture components like the Feed-Forward Network (semantic refinement) and Multi-Head Attention (context integration).

Empirical Validation

Empirical analyses across Transformer architectures validate DMET's propositions. Key experiments involved the evaluation of three LLMs: DeepSeek-R1, Llama2, and Qwen2, using prompts that varied in complexity. The experiments leveraged a grid of temperature and top-$p$ values to assess how these parameters affect latent trajectories and text quality.

Findings:

• Continuity (C) was found to correlate with lower perplexity and improved fluency.
• Attractor Compactness (Q) was associated with grammatical consistency.
• Topological Persistence (P) was a primary predictor of semantic coherence.

  Figure 2: Latent representations reside on a low-dimensional semantic manifold.

Trajectory Dynamics and Parameter Sensitivity

The study explores how decoding parameters shape latent trajectories, revealing that:

• Low temperature enhances continuity but reduces diversity.
• High top-$p$ values allow for broader exploration, beneficial for creative tasks but potentially at the cost of fluency.

The findings support DMET's predictions and illustrate a trade-off between fluency and creativity across the parameter space.

Figure 3: Fluency–Coherence trade-off under varying temperature and top-$p$ using DeepSeek-R1.

Practical Implications and Limitations

The study identifies actionable strategies for parameter tuning, such as using moderate temperatures and top-$p$ values for balanced generation outcomes. However, limitations include the high computational complexity of manifold analyses and the need to establish causal links between latent dynamics and text quality.

Conclusions

This work establishes DMET as a robust framework for interpreting LLM behavior by relating latent dynamics to text quality. Future work could focus on developing efficient algorithms for real-time manifold analyses and exploring direct interventions to manipulate latent trajectories for improved control over LLM outputs. The insights provided by DMET offer a principled basis for designing and understanding next-generation LLMs.