# The Linear Representation Hypothesis and the Geometry of Large Language Models

arXiv: https://arxiv.org/abs/2311.03658 · HTML: https://mlsanity.com/paper/2311.03658/

## Key facts

- Cited by 716 papers in the map (data through 14 Oct 2026).
- Builds on 43 of its 65 references that are in the map.
- First posted on arXiv 7 Nov 2023.

## Builds on (most cited first, 30 of 43)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [GPT-4 Technical Report](https://mlsanity.com/paper/2303.08774/) (2023)
- [Llama 2: Open Foundation and Fine-Tuned Chat Models](https://mlsanity.com/paper/2307.09288/) (2023)
- [Distributed Representations of Words and Phrases and their Compositionality](https://mlsanity.com/paper/1310.4546/) (2013)
- [Locating and Editing Factual Associations in GPT](https://mlsanity.com/paper/2202.05262/) (2022)
- [SentencePiece: A simple and language independent subword tokenizer and detokenizer for Neural Text Processing](https://mlsanity.com/paper/1808.06226/) (2018)
- [Understanding intermediate layers using linear classifier probes](https://mlsanity.com/paper/1610.01644/) (2016)
- [Representation Engineering: A Top-Down Approach to AI Transparency](https://mlsanity.com/paper/2310.01405/) (2023)
- [Interpretability Beyond Feature Attribution: Quantitative Testing with Concept Activation Vectors (TCAV)](https://mlsanity.com/paper/1711.11279/) (2017)
- [Toy Models of Superposition](https://mlsanity.com/paper/2209.10652/) (2022)
- [Steering Language Models With Activation Engineering](https://mlsanity.com/paper/2308.10248/) (2023)
- [Probing Classifiers: Promises, Shortcomings, and Advances](https://mlsanity.com/paper/2102.12452/) (2021)
- [A Primer in BERTology: What we know about how BERT works](https://mlsanity.com/paper/2002.12327/) (2020)
- [How Contextual are Contextualized Word Representations? Comparing the Geometry of BERT, ELMo, and GPT-2 Embeddings](https://mlsanity.com/paper/1909.00512/) (2019)
- [Generating Sentences from a Continuous Space](https://mlsanity.com/paper/1511.06349/) (2015)
- [Transformer Feed-Forward Layers Build Predictions by Promoting Concepts in the Vocabulary Space](https://mlsanity.com/paper/2203.14680/) (2022)
- [Variational Autoencoders and Nonlinear ICA: A Unifying Framework](https://mlsanity.com/paper/1907.04809/) (2019)
- [Word Translation Without Parallel Data](https://mlsanity.com/paper/1710.04087/) (2017)
- [Emergent World Representations: Exploring a Sequence Model Trained on a Synthetic Task](https://mlsanity.com/paper/2210.13382/) (2022)
- [Emergent Linear Representations in World Models of Self-Supervised Sequence Models](https://mlsanity.com/paper/2309.00941/) (2023)
- [Language Models Represent Space and Time](https://mlsanity.com/paper/2310.02207/) (2023)
- [In-Context Learning Creates Task Vectors](https://mlsanity.com/paper/2310.15916/) (2023)
- [Function Vectors in Large Language Models](https://mlsanity.com/paper/2310.15213/) (2023)
- [Towards a Definition of Disentangled Representations](https://mlsanity.com/paper/1812.02230/) (2018)
- [Unsupervised Feature Extraction by Time-Contrastive Learning and Nonlinear ICA](https://mlsanity.com/paper/1605.06336/) (2016)
- [Contrastive Learning Inverts the Data Generating Process](https://mlsanity.com/paper/2102.08850/) (2021)
- [Visualizing and Measuring the Geometry of BERT](https://mlsanity.com/paper/1906.02715/) (2019)
- [Linearity of Relation Decoding in Transformer Language Models](https://mlsanity.com/paper/2308.09124/) (2023)
- [Exploiting Similarities among Languages for Machine Translation](https://mlsanity.com/paper/1309.4168/) (2013)
- [A Survey Of Cross-lingual Word Embedding Models](https://mlsanity.com/paper/1706.04902/) (2017)

## Built on by (most cited first, 30 of 716)

- [Refusal in Language Models Is Mediated by a Single Direction](https://mlsanity.com/paper/2406.11717/) (2024)
- [Mechanistic Interpretability for AI Safety -- A Review](https://mlsanity.com/paper/2404.14082/) (2024)
- [Layer by Layer: Uncovering Hidden Representations in Language Models](https://mlsanity.com/paper/2502.02013/) (2025)
- [AxBench: Steering LLMs? Even Simple Baselines Outperform Sparse Autoencoders](https://mlsanity.com/paper/2501.17148/) (2025)
- [Not All Language Model Features Are One-Dimensionally Linear](https://mlsanity.com/paper/2405.14860/) (2024)
- [Programming Refusal with Conditional Activation Steering](https://mlsanity.com/paper/2409.05907/) (2024)
- [ReFT: Representation Finetuning for Language Models](https://mlsanity.com/paper/2404.03592/) (2024)
- [A Mechanistic Understanding of Alignment Algorithms: A Case Study on DPO and Toxicity](https://mlsanity.com/paper/2401.01967/) (2024)
- [Improving Dictionary Learning with Gated Sparse Autoencoders](https://mlsanity.com/paper/2404.16014/) (2024)
- [A is for Absorption: Studying Feature Splitting and Absorption in Sparse Autoencoders](https://mlsanity.com/paper/2409.14507/) (2024)
- [Improving Instruction-Following in Language Models through Activation Steering](https://mlsanity.com/paper/2410.12877/) (2024)
- [The Geometry of Categorical and Hierarchical Concepts in Large Language Models](https://mlsanity.com/paper/2406.01506/) (2024)
- [Analyzing the Generalization and Reliability of Steering Vectors](https://mlsanity.com/paper/2407.12404/) (2024)
- [Interpreting CLIP with Sparse Linear Concept Embeddings (SpLiCE)](https://mlsanity.com/paper/2402.10376/) (2024)
- [Do I Know This Entity? Knowledge Awareness and Hallucinations in Language Models](https://mlsanity.com/paper/2411.14257/) (2024)
- [The Geometry of Refusal in Large Language Models: Concept Cones and Representational Independence](https://mlsanity.com/paper/2502.17420/) (2025)
- [Automatically Interpreting Millions of Features in Large Language Models](https://mlsanity.com/paper/2410.13928/) (2024)
- [LLM Social Simulations Are a Promising Research Method](https://mlsanity.com/paper/2504.02234/) (2025)
- [Model Organisms for Emergent Misalignment](https://mlsanity.com/paper/2506.11613/) (2025)
- [A Primer on the Inner Workings of Transformer-based Language Models](https://mlsanity.com/paper/2405.00208/) (2024)
- [Verbalizable Representations Form a Global Workspace in Language Models](https://mlsanity.com/paper/2607.15495/) (2026)
- [Latent Collaboration in Multi-Agent Systems](https://mlsanity.com/paper/2511.20639/) (2025)
- [Convergent Linear Representations of Emergent Misalignment](https://mlsanity.com/paper/2506.11618/) (2025)
- [Attention Heads of Large Language Models: A Survey](https://mlsanity.com/paper/2409.03752/) (2024)
- [Unintentional Unalignment: Likelihood Displacement in Direct Preference Optimization](https://mlsanity.com/paper/2410.08847/) (2024)
- [Robust LLM safeguarding via refusal feature adversarial training](https://mlsanity.com/paper/2409.20089/) (2024)
- [Semantics-Adaptive Activation Intervention for LLMs via Dynamic Steering Vectors](https://mlsanity.com/paper/2410.12299/) (2024)
- [JBShield: Defending Large Language Models from Jailbreak Attacks through Activated Concept Analysis and Manipulation](https://mlsanity.com/paper/2502.07557/) (2025)
- [Angular Steering: Behavior Control via Rotation in Activation Space](https://mlsanity.com/paper/2510.26243/) (2025)
- [The Geometry of Concepts: Sparse Autoencoder Feature Structure](https://mlsanity.com/paper/2410.19750/) (2024)

## Abstract (arXiv)

Informally, the 'linear representation hypothesis' is the idea that high-level concepts are represented linearly as directions in some representation space. In this paper, we address two closely related questions: What does "linear representation" actually mean? And, how do we make sense of geometric notions (e.g., cosine similarity or projection) in the representation space? To answer these, we use the language of counterfactuals to give two formalizations of "linear representation", one in the output (word) representation space, and one in the input (sentence) space. We then prove these connect to linear probing and model steering, respectively. To make sense of geometric notions, we use the formalization to identify a particular (non-Euclidean) inner product that respects language structure in a sense we make precise. Using this causal inner product, we show how to unify all notions of linear representation. In particular, this allows the construction of probes and steering vectors using counterfactual pairs. Experiments with LLaMA-2 demonstrate the existence of linear representations of concepts, the connection to interpretation and control, and the fundamental role of the choice of inner product.