# Toward a Flexible Framework for Linear Representation Hypothesis Using Maximum Likelihood Estimation

arXiv: https://arxiv.org/abs/2502.16385 · HTML: https://mlsanity.com/paper/2502.16385/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 33 of its 39 references that are in the map.
- First posted on arXiv 22 Feb 2025.

## Builds on (most cited first, 30 of 33)

- [GPT-4 Technical Report](https://mlsanity.com/paper/2303.08774/) (2023)
- [Think you have Solved Question Answering? Try ARC, the AI2 Reasoning Challenge](https://mlsanity.com/paper/1803.05457/) (2018)
- [Can a Suit of Armor Conduct Electricity? A New Dataset for Open Book Question Answering](https://mlsanity.com/paper/1809.02789/) (2018)
- [CommonsenseQA: A Question Answering Challenge Targeting Commonsense Knowledge](https://mlsanity.com/paper/1811.00937/) (2018)
- [Representation Engineering: A Top-Down Approach to AI Transparency](https://mlsanity.com/paper/2310.01405/) (2023)
- [Inference-Time Intervention: Eliciting Truthful Answers from a Language Model](https://mlsanity.com/paper/2306.03341/) (2023)
- [Steering Llama 2 via Contrastive Activation Addition](https://mlsanity.com/paper/2312.06681/) (2023)
- [Refusal in Language Models Is Mediated by a Single Direction](https://mlsanity.com/paper/2406.11717/) (2024)
- [Steering Language Models With Activation Engineering](https://mlsanity.com/paper/2308.10248/) (2023)
- [RACE: Large-scale ReAding Comprehension Dataset From Examinations](https://mlsanity.com/paper/1704.04683/) (2017)
- [Discovering Latent Knowledge in Language Models Without Supervision](https://mlsanity.com/paper/2212.03827/) (2022)
- [Aligning AI With Shared Human Values](https://mlsanity.com/paper/2008.02275/) (2020)
- [The Linear Representation Hypothesis and the Geometry of Large Language Models](https://mlsanity.com/paper/2311.03658/) (2023)
- [The Geometry of Truth: Emergent Linear Structure in Large Language Model Representations of True/False Datasets](https://mlsanity.com/paper/2310.06824/) (2023)
- [Language Models Represent Space and Time](https://mlsanity.com/paper/2310.02207/) (2023)
- [In-context Vectors: Making In Context Learning More Effective and Controllable Through Latent Space Steering](https://mlsanity.com/paper/2311.06668/) (2023)
- [Extracting Latent Steering Vectors from Pretrained Language Models](https://mlsanity.com/paper/2205.05124/) (2022)
- [Linear Representations of Sentiment in Large Language Models](https://mlsanity.com/paper/2310.15154/) (2023)
- [The Geometry of Categorical and Hierarchical Concepts in Large Language Models](https://mlsanity.com/paper/2406.01506/) (2024)
- [Personalized Steering of Large Language Models: Versatile Steering Vectors Through Bi-directional Preference Optimization](https://mlsanity.com/paper/2406.00045/) (2024)
- [Steering Language Model Refusal with Sparse Autoencoders](https://mlsanity.com/paper/2411.11296/) (2024)
- [On the Origins of Linear Representations in Large Language Models](https://mlsanity.com/paper/2403.03867/) (2024)
- [Improving Activation Steering in Language Models with Mean-Centring](https://mlsanity.com/paper/2312.03813/) (2023)
- [Anisotropy Is Inherent to Self-Attention in Transformers](https://mlsanity.com/paper/2401.12143/) (2024)
- [Steering Knowledge Selection Behaviours in LLMs via SAE-Based Representation Engineering](https://mlsanity.com/paper/2410.15999/) (2024)
- [Eliciting Latent Knowledge from Quirky Language Models](https://mlsanity.com/paper/2312.01037/) (2023)
- [Representation Surgery: Theory and Practice of Affine Steering](https://mlsanity.com/paper/2402.09631/) (2024)
- [A Language Model's Guide Through Latent Space](https://mlsanity.com/paper/2402.14433/) (2024)
- [The Shape of Learning: Anisotropy and Intrinsic Dimensions in Transformer-Based Models](https://mlsanity.com/paper/2311.05928/) (2023)
- [Steering Large Language Models using Conceptors: Improving Addition-Based Activation Engineering](https://mlsanity.com/paper/2410.16314/) (2024)

## Built on by (most cited first, 3 of 3)

- [The Cylindrical Representation Hypothesis for Language Model Steering](https://mlsanity.com/paper/2605.01844/) (2026)
- [Patches of Nonlinearity: Instruction Vectors in Large Language Models](https://mlsanity.com/paper/2602.07930/) (2026)
- [Mechanistic Interpretability for Neural Networks: Circuits, Sparse Features and Symbolic Reasoning](https://mlsanity.com/paper/2607.07316/) (2026)

## Abstract (arXiv)

Linear representation hypothesis posits that high-level concepts are encoded as linear directions in the representation spaces of LLMs. Park et al. (2024) formalize this notion by unifying multiple interpretations of linear representation, such as 1-dimensional subspace representation and interventions, using a causal inner product. However, their framework relies on single-token counterfactual pairs and cannot handle ambiguous contrasting pairs, limiting its applicability to complex or context-dependent concepts. We introduce a new notion of binary concepts as unit vectors in a canonical representation space, and utilize LLMs' (neural) activation differences along with maximum likelihood estimation (MLE) to compute concept directions (i.e., steering vectors). Our method, Sum of Activation-base Normalized Difference (SAND), formalizes the use of activation differences modeled as samples from a von Mises-Fisher (vMF) distribution, providing a principled approach to derive concept directions. We extend the applicability of Park et al. (2024) by eliminating the dependency on unembedding representations and single-token pairs. Through experiments with LLaMA models across diverse concepts and benchmarks, we demonstrate that our lightweight approach offers greater flexibility, superior performance in activation engineering tasks like monitoring and manipulation.