# Language Models Represent and Transform Concepts with Shared Geometry (Raw Archive)

- **arXiv**: 2607.04525v1
- **Authors**: Zhimin Hu (Georgia Tech), Lanhao Niu (U. Edinburgh), Sashank Varma (Georgia Tech)
- **Published**: July 5, 2026
- **Venue**: ICML 2026 (Proceedings of the 43rd ICML, Seoul)
- **Category**: cs.CL, cs.AI

## Abstract

How concepts are represented in neural networks is a fundamental question. The dominant view treats concept representations as stationary geometric objects. Yet concepts appear in context, and context transforms them. Drawing from neural population geometry, we formalize concept representations as point-cloud manifolds and contextual transformations as vector fields, and instantiate this framework in large language models. Across six model families of varying scales, we find that context moves each concept differently. The variance in these displacements is semantically organized, correlating with lexical concreteness and density. Importantly, both the concepts being transformed and this variance structure are shared across models: displacement structure transported from one model predicts held-out displacements in others significantly above chance.

## Key Framework

- **Concepts as Point-Cloud Manifolds**: X_w(m) = {r(m)(w, c_i)} — each concept is an empirical sample from an underlying distribution
- **Two Axes of Representation**: Within-concept (shape of point cloud across contexts) and Between-concept (kernel matrix of concepts in shared context)
- **Contextual Transformation Field**: φ(m)(w, τ) = r(m)(w, τ) - r(m)(w, τ_0), characterized by its kernel matrix
- **Cross-Model Measurement**: CKA for relational structure; Grassmann distance for directional alignment; permutation calibration for significance
- **23 models across 6 families** (0.5B–32B): Qwen2.5, Llama, Mistral, Gemma2, DeepSeek, Pythia
