Geometric Analysis of Reasoning Trajectories: A Phase Space Approach to Understanding Valid and Invalid Multi-Hop Reasoning in LLMs

TL;DR

This paper introduces a physics-inspired geometric framework using Hamiltonian mechanics to analyze and distinguish valid from invalid multi-hop reasoning in large language models, providing new diagnostic tools.

Contribution

It presents a novel phase space approach to visualize and quantify reasoning trajectories in LLMs, revealing geometric patterns associated with reasoning validity.

Findings

Valid reasoning exhibits lower Hamiltonian energy values.

The framework enables visualization of reasoning trajectories in embedding space.

Distinct geometric patterns differentiate valid from invalid reasoning.

Abstract

This paper proposes a novel approach to analyzing multi-hop reasoning in language models through Hamiltonian mechanics. We map reasoning chains in embedding spaces to Hamiltonian systems, defining a function that balances reasoning progression (kinetic energy) against question relevance (potential energy). Analyzing reasoning chains from a question-answering dataset reveals that valid reasoning shows lower Hamiltonian energy values, representing an optimal trade-off between information gathering and targeted answering. While our framework offers complex visualization and quantification methods, the claimed ability to "steer" or "improve" reasoning algorithms requires more rigorous empirical validation, as the connection between physical systems and reasoning remains largely metaphorical. Nevertheless, our analysis reveals consistent geometric patterns distinguishing valid reasoning, suggesting this physics-inspired approach offers promising diagnostic tools and new perspectives on reasoning processes in large language models.