# Shared Global and Local Geometry of Language Model Embeddings

arXiv: https://arxiv.org/abs/2503.21073 · HTML: https://mlsanity.com/paper/2503.21073/

## Key facts

- Cited by 36 papers in the map (data through 14 Oct 2026).
- Builds on 42 of its 52 references that are in the map.
- First posted on arXiv 27 Mar 2025.

## Builds on (most cited first, 30 of 42)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [The Llama 3 Herd of Models](https://mlsanity.com/paper/2407.21783/) (2024)
- [Distributed Representations of Words and Phrases and their Compositionality](https://mlsanity.com/paper/1310.4546/) (2013)
- [The Pile: An 800GB Dataset of Diverse Text for Language Modeling](https://mlsanity.com/paper/2101.00027/) (2020)
- [Gemma 2: Improving Open Language Models at a Practical Size](https://mlsanity.com/paper/2408.00118/) (2024)
- [Inference-Time Intervention: Eliciting Truthful Answers from a Language Model](https://mlsanity.com/paper/2306.03341/) (2023)
- [RealToxicityPrompts: Evaluating Neural Toxic Degeneration in Language Models](https://mlsanity.com/paper/2009.11462/) (2020)
- [Steering Llama 2 via Contrastive Activation Addition](https://mlsanity.com/paper/2312.06681/) (2023)
- [ConceptNet 5.5: An Open Multilingual Graph of General Knowledge](https://mlsanity.com/paper/1612.03975/) (2016)
- [Discovering Language Model Behaviors with Model-Written Evaluations](https://mlsanity.com/paper/2212.09251/) (2022)
- [Prevalence of Neural Collapse during the terminal phase of deep learning training](https://mlsanity.com/paper/2008.08186/) (2020)
- [The Linear Representation Hypothesis and the Geometry of Large Language Models](https://mlsanity.com/paper/2311.03658/) (2023)
- [GPT-NeoX-20B: An Open-Source Autoregressive Language Model](https://mlsanity.com/paper/2204.06745/) (2022)
- [OLMo: Accelerating the Science of Language Models](https://mlsanity.com/paper/2402.00838/) (2024)
- [Transformer Feed-Forward Layers Build Predictions by Promoting Concepts in the Vocabulary Space](https://mlsanity.com/paper/2203.14680/) (2022)
- [Dolma: an Open Corpus of Three Trillion Tokens for Language Model Pretraining Research](https://mlsanity.com/paper/2402.00159/) (2024)
- [The Platonic Representation Hypothesis](https://mlsanity.com/paper/2405.07987/) (2024)
- [Emergent Linear Representations in World Models of Self-Supervised Sequence Models](https://mlsanity.com/paper/2309.00941/) (2023)
- [Understanding image representations by measuring their equivariance and equivalence](https://mlsanity.com/paper/1411.5908/) (2014)
- [Contrastive Learning Inverts the Data Generating Process](https://mlsanity.com/paper/2102.08850/) (2021)
- [Revisiting Model Stitching to Compare Neural Representations](https://mlsanity.com/paper/2106.07682/) (2021)
- [Exploiting Similarities among Languages for Machine Translation](https://mlsanity.com/paper/1309.4168/) (2013)
- [A Mechanistic Understanding of Alignment Algorithms: A Case Study on DPO and Toxicity](https://mlsanity.com/paper/2401.01967/) (2024)
- [Emerging Cross-lingual Structure in Pretrained Language Models](https://mlsanity.com/paper/1911.01464/) (2019)
- [The Geometry of Categorical and Hierarchical Concepts in Large Language Models](https://mlsanity.com/paper/2406.01506/) (2024)
- [A Toy Model of Universality: Reverse Engineering How Networks Learn Group Operations](https://mlsanity.com/paper/2302.03025/) (2023)
- [Linearly Mapping from Image to Text Space](https://mlsanity.com/paper/2209.15162/) (2022)
- [Gromov-Wasserstein Alignment of Word Embedding Spaces](https://mlsanity.com/paper/1809.00013/) (2018)
- [Adversarial Reprogramming of Neural Networks](https://mlsanity.com/paper/1806.11146/) (2018)
- [The geometry of hidden representations of large transformer models](https://mlsanity.com/paper/2302.00294/) (2023)

## Built on by (most cited first, 30 of 36)

- [Transferring Linear Features Across Language Models With Model Stitching](https://mlsanity.com/paper/2506.06609/) (2025)
- [Token embeddings violate the manifold hypothesis](https://mlsanity.com/paper/2504.01002/) (2025)
- [Valence-Arousal Subspace in LLMs: Circular Emotion Geometry and Multi-Behavioral Control](https://mlsanity.com/paper/2604.03147/) (2026)
- [The Geometry of Self-Verification in a Task-Specific Reasoning Model](https://mlsanity.com/paper/2504.14379/) (2025)
- [Do different prompting methods yield a common task representation in language models?](https://mlsanity.com/paper/2505.12075/) (2025)
- [Representation Potentials of Foundation Models for Multimodal Alignment: A Survey](https://mlsanity.com/paper/2510.05184/) (2025)
- [From Data Statistics to Feature Geometry: How Correlations Shape Superposition](https://mlsanity.com/paper/2603.09972/) (2026)
- [Differential syntactic and semantic encoding in LLMs](https://mlsanity.com/paper/2601.04765/) (2026)
- [Learning Through Noise: Why Subliminal Learning Works and When It Fails](https://mlsanity.com/paper/2605.23645/) (2026)
- [Jailbreak Transferability Emerges from Shared Representations](https://mlsanity.com/paper/2506.12913/) (2025)
- [From Topology to Retrieval: Decoding Embedding Spaces with Unified Signatures](https://mlsanity.com/paper/2511.22150/) (2025)
- [Signal in the Noise: Polysemantic Interference Transfers and Predicts Cross-Model Influence](https://mlsanity.com/paper/2505.11611/) (2025)
- [Training-Free Tokenizer Transplantation via Orthogonal Matching Pursuit](https://mlsanity.com/paper/2506.06607/) (2025)
- [Better World Models Can Lead to Better Post-Training Performance](https://mlsanity.com/paper/2512.03400/) (2025)
- [GeoSteer: Faithful Chain-of-Thought Steering via Latent Manifold Gradients](https://mlsanity.com/paper/2601.10229/) (2026)
- [Zipage: Maintain High Request Concurrency for LLM Reasoning through Compressed PagedAttention](https://mlsanity.com/paper/2603.08743/) (2026)
- [The Indra Representation Hypothesis for Multimodal Alignment](https://mlsanity.com/paper/2604.04496/) (2026)
- [Latent Structure of Affective Representations in Large Language Models](https://mlsanity.com/paper/2604.07382/) (2026)
- [Disentangling MLP Neuron Weights in Vocabulary Space](https://mlsanity.com/paper/2604.06005/) (2026)
- [Position: Require Frontier AI Labs To Release Small "Analog" Models](https://mlsanity.com/paper/2510.14053/) (2025)
- [Abstraction Induces the Brain Alignment of Language and Speech Models](https://mlsanity.com/paper/2602.04081/) (2026)
- [Characterizing Human Semantic Navigation in Concept Production as Trajectories in Embedding Space](https://mlsanity.com/paper/2602.05971/) (2026)
- [Scale Determines Whether Language Models Organize Representation Geometry for Prediction](https://mlsanity.com/paper/2605.17084/) (2026)
- [Decoding in Geometry: Alleviating Embedding-Space Crowding for Complex Reasoning](https://mlsanity.com/paper/2601.22536/) (2026)
- [What Cosine Similarity of Label Representations Can and Cannot Tell us](https://mlsanity.com/paper/2603.29488/) (2026)
- [Tensor Product Representation Probes Reveal Shared Structure Across Linear Directions](https://mlsanity.com/paper/2605.09967/) (2026)
- [DreamReader: An Interpretability Toolkit for Text-to-Image Models](https://mlsanity.com/paper/2603.13299/) (2026)
- [The Cost of Language: Centroid Erasure Exposes and Exploits Modal Competition in Multimodal Language Models](https://mlsanity.com/paper/2604.14363/) (2026)
- [RDP LoRA: Geometry-Driven Identification for Parameter-Efficient Adaptation in Large Language Models](https://mlsanity.com/paper/2604.19321/) (2026)
- [Coordinates of Capability: A Unified MTMM-Geometric Framework for LLM Evaluation](https://mlsanity.com/paper/2605.08522/) (2026)

## Abstract (arXiv)

Researchers have recently suggested that models share common representations. In our work, we find numerous geometric similarities across the token embeddings of large language models. First, we find ``global'' similarities: token embeddings often share similar relative orientations. Next, we characterize local geometry in two ways: (1) by using Locally Linear Embeddings, and (2) by defining a simple measure for the intrinsic dimension of each embedding. Both characterizations allow us to find local similarities across token embeddings. Additionally, our intrinsic dimension demonstrates that embeddings lie on a lower dimensional manifold, and that tokens with lower intrinsic dimensions often have semantically coherent clusters, while those with higher intrinsic dimensions do not. Based on our findings, we introduce EMB2EMB, a simple application to linearly transform steering vectors from one language model to another, despite the two models having different dimensions.