The Geometry of Tokens in Internal Representations of Large Language Models

arXiv:2501.10573 · cs.CL, cs.LG · Submitted 2026-08-21 · Read on arXiv

Karthik Viswanathan, Yuri Gardinazzi, Giada Panerai, Alberto Cazzaniga, Matteo Biagetti

cs.CL, cs.LG

Submitted: 2026-08-21

Updated: 2026-08-24

Comments: 12+14 pages, 18 figures, matches published version on Transactions of Machine Learning Research

Journal ref: Transactions of Machine Learning Research, ISSN: 2835-8856, 2026, https://openreview.net/pdf?id=rBEgNAslpY

Code: https://github.com/RitAreaSciencePark/token

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Terminology

• Intrinsic Dimension — 57× in this paper · explained in 2 episodes
• Transformer — 41× in this paper · explained in 2 episodes
• Cosine Similarity — 21× in this paper · explained in 3 episodes
• Transformer Models — 11× in this paper · explained in 2 episodes
• Interpretability — 7× in this paper · explained in 15 episodes
• Clustering — 5× in this paper · explained in 2 episodes
• Consistency — 5× in this paper · explained in 3 episodes
• Embeddings — 4× in this paper · explained in 2 episodes
• Point Clouds — 3× in this paper · explained in 2 episodes
• Abstraction — 2× in this paper · explained in 1 episode
• Alignment — 2× in this paper · explained in 1 episode
• Linear probe — 2× in this paper · explained in 2 episodes
• Metastability — 2× in this paper · explained in 2 episodes
• Perplexity — 2× in this paper · explained in 3 episodes
• Transformer Architecture — 2× in this paper · explained in 2 episodes
• Accuracy — 1× in this paper · explained in 1 episode
• Activation Function — 1× in this paper · explained in 1 episode
• Anisotropy — 1× in this paper · explained in 3 episodes
• Conference on Computational Natural Language Learning — 1× in this paper
• Expected Simplex Skewness — 1× in this paper
• Few-Shot Learning — 1× in this paper · explained in 4 episodes
• Fine-Tuning — 1× in this paper · explained in 5 episodes
• Homogeneity — 1× in this paper · explained in 2 episodes
• In-Context Learning — 1× in this paper · explained in 4 episodes
• Modality Gap — 1× in this paper · explained in 4 episodes
• Over-smoothing — 1× in this paper · explained in 4 episodes
• Representation Learning — 1× in this paper · explained in 2 episodes
• Sparsity — 1× in this paper · explained in 4 episodes
• Tokenization — 1× in this paper · explained in 2 episodes

Sources

• A Mathematical Theory of Attention
• A mathematical perspective on Transformers
• Geometric Dynamics of Signal Propagation Predict Trainability of Transformers
• Generic controllability of equivariant systems and applications to particle systems and neural networks
• Eliciting Latent Predictions from Transformers with the Tuned Lens
• Emergence of a High-Dimensional Abstraction Phase in Language Transformers
• Persistent Topological Features in Large Language Models
• Levels of Analysis for Machine Learning
• Measure-to-measure interpolation using Transformers
• Dynamic metastability in the self-attention model
• Unsupervised detection of semantic correlations in big data
• Implicit Geometry of Next-token Prediction: From Language Sparsity Patterns to Model Representations
• Distributional Results for Model-Based Intrinsic Dimension Estimators
• Mistral 7B
• The Pile: An 800GB Dataset of Diverse Text for Language Modeling
• Loss Landscape Degeneracy and Stagewise Development in Transformers
• Lines of Thought in Large Language Models

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