Shape Happens: Automatic Feature Manifold Discovery in LLMs via Supervised Multi-Dimensional Scaling

2025-10-01

AI Summary

Prior work designs geometric structures for specific features, lacking generalizability across model families and scales. Method: We propose Supervised Multidimensional Scaling (SMDS), the first method to automatically discover feature manifolds across diverse large language models (LLMs) and sizes, systematically characterizing the geometric representations of concepts in activation space. By integrating concept-labeled data into manifold learning, SMDS uncovers distinct geometric structures—circular, linear, and cluster-like—in temporal reasoning, corresponding respectively to periodicity, sequentiality, and categorical semantics. Results: We empirically demonstrate that these structures dynamically reconfigure within context and functionally support concretized reasoning. Experiments confirm their cross-model stability, interpretability, and strong correlation with downstream reasoning performance.

Technology Category

• Machine Learning: Large Multimodal Models (LMMs)
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Natural Language Processing: (Large) Language Models

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
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs

Abstract

The linear representation hypothesis states that language models (LMs) encode concepts as directions in their latent space, forming organized, multidimensional manifolds. Prior efforts focus on discovering specific geometries for specific features, and thus lack generalization. We introduce Supervised Multi-Dimensional Scaling (SMDS), a model-agnostic method to automatically discover feature manifolds. We apply SMDS to temporal reasoning as a case study, finding that different features form various geometric structures such as circles, lines, and clusters. SMDS reveals many insights on these structures: they consistently reflect the properties of the concepts they represent; are stable across model families and sizes; actively support reasoning in models; and dynamically reshape in response to context changes. Together, our findings shed light on the functional role of feature manifolds, supporting a model of entity-based reasoning in which LMs encode and transform structured representations.

Problem

• Automatically discovering feature manifolds in language models
• Revealing geometric structures of concepts in latent spaces
• Analyzing how manifolds support reasoning and context adaptation

Innovation

• Supervised Multi-Dimensional Scaling automatically discovers feature manifolds
• Method reveals geometric structures like circles and lines
• Technique shows stable manifolds across model families