The Geometry of Knowledge Accessibility in Large Language Models

2026-10-02

AI Summary

This study addresses the unreliability of knowledge access in large language models and the difficulty of existing methods in disentangling knowledge boundaries from reasoning complexity. We propose a geometric theory of knowledge access grounded in representational space distance, revealing that proximity to the center of the query representation space correlates with higher accessibility. Based on this insight, we construct a pre-generation predictive signal to guide adaptive reasoning interventions, including rewriting, chain-of-thought prompting, and retrieval augmentation. Our contributions demonstrate the cross-dataset transferability of this distance-based ranking mechanism and clarify the differential effectiveness of various intervention strategies within and beyond knowledge boundaries, thereby establishing a novel paradigm for efficient adaptive reasoning.

Abstract

Large language models (LLMs) contain broad knowledge, but they cannot access all of it reliably. We study this problem through knowledge accessibility, which describes whether the knowledge needed for a query can be recalled from the model. We find that knowledge accessibility has a simple geometric structure in the model's representation of the query alone, before any generation. More accessible queries are closer to a center in the representation space, while less accessible queries are farther away. This geometry reveals a knowledge boundary that separates more accessible queries from less accessible ones. Accessibility consistently decreases with distance from the center, and this distance-based ordering transfers across datasets even when the centers differ. Controlled experiments further show that the centered geometry is more closely related to knowledge accessibility than to reasoning difficulty. The geometry also reveals when different interventions are useful. Query rewriting helps more for accessible queries, chain-of-thought reasoning helps more near the boundary, and retrieval gives larger gains beyond the boundary. These findings not only provide a new geometric view of how knowledge is organized in language models, but also suggest a useful pre-generation signal for adaptive inference.

Problem

• Large Language Models
• Knowledge Accessibility
• Representation Geometry
• Knowledge Boundary

Innovation

• Knowledge Accessibility
• Geometric Structure
• Large Language Models
• Knowledge Boundary
• Adaptive Inference