# The Geometry of Concepts: Sparse Autoencoder Feature Structure

arXiv: https://arxiv.org/abs/2410.19750 · HTML: https://mlsanity.com/paper/2410.19750/

## Key facts

- Cited by 52 papers in the map (data through 14 Oct 2026).
- Placed in Sparse autoencoder training and evaluation, one of 13 directions in Mechanistic interpretability.
- Builds on 28 of its 70 references that are in the map.
- First posted on arXiv 10 Oct 2024.

## Placed in

- Mechanistic interpretability / [Sparse autoencoder training and evaluation](https://mlsanity.com/mechinterp/sparse-autoencoder-training-and-evaluation/)

## Builds on (most cited first, 28 of 28)

- [Locating and Editing Factual Associations in GPT](https://mlsanity.com/paper/2202.05262/) (2022)
- [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)
- [Sparse Autoencoders Find Highly Interpretable Features in Language Models](https://mlsanity.com/paper/2309.08600/) (2023)
- [Towards Understanding Sycophancy in Language Models](https://mlsanity.com/paper/2310.13548/) (2023)
- [Toy Models of Superposition](https://mlsanity.com/paper/2209.10652/) (2022)
- [Towards Automated Circuit Discovery for Mechanistic Interpretability](https://mlsanity.com/paper/2304.14997/) (2023)
- [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)
- [Eliciting Latent Predictions from Transformers with the Tuned Lens](https://mlsanity.com/paper/2303.08112/) (2023)
- [Emergent World Representations: Exploring a Sequence Model Trained on a Synthetic Task](https://mlsanity.com/paper/2210.13382/) (2022)
- [Mechanistic Interpretability for AI Safety -- A Review](https://mlsanity.com/paper/2404.14082/) (2024)
- [Gemma Scope: Open Sparse Autoencoders Everywhere All At Once on Gemma 2](https://mlsanity.com/paper/2408.05147/) (2024)
- [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)
- [Jumping Ahead: Improving Reconstruction Fidelity with JumpReLU Sparse Autoencoders](https://mlsanity.com/paper/2407.14435/) (2024)
- [Intrinsic dimension of data representations in deep neural networks](https://mlsanity.com/paper/1905.12784/) (2019)
- [The Alignment Problem from a Deep Learning Perspective](https://mlsanity.com/paper/2209.00626/) (2022)
- [AI Deception: A Survey of Examples, Risks, and Potential Solutions](https://mlsanity.com/paper/2308.14752/) (2023)
- [The Geometry of Categorical and Hierarchical Concepts in Large Language Models](https://mlsanity.com/paper/2406.01506/) (2024)
- [Transformer visualization via dictionary learning: contextualized embedding as a linear superposition of transformer factors](https://mlsanity.com/paper/2103.15949/) (2021)
- [Evolution of Semantic Similarity -- A Survey](https://mlsanity.com/paper/2004.13820/) (2020)
- [Sparse Overcomplete Word Vector Representations](https://mlsanity.com/paper/1506.02004/) (2015)
- [Explorations of Self-Repair in Language Models](https://mlsanity.com/paper/2402.15390/) (2024)
- [Monotonic Representation of Numeric Properties in Language Models](https://mlsanity.com/paper/2403.10381/) (2024)
- [Word Embedding Visualization Via Dictionary Learning](https://mlsanity.com/paper/1910.03833/) (2019)

## Built on by (most cited first, 30 of 52)

- [Understanding World or Predicting Future? A Comprehensive Survey of World Models](https://mlsanity.com/paper/2411.14499/) (2024)
- [Are Sparse Autoencoders Useful? A Case Study in Sparse Probing](https://mlsanity.com/paper/2502.16681/) (2025)
- [The Origins of Representation Manifolds in Large Language Models](https://mlsanity.com/paper/2505.18235/) (2025)
- [When Models Manipulate Manifolds: The Geometry of a Counting Task](https://mlsanity.com/paper/2601.04480/) (2026)
- [From Tokens to Thoughts: How LLMs and Humans Trade Compression for Meaning](https://mlsanity.com/paper/2505.17117/) (2025)
- [Beyond Prompt Engineering: Robust Behavior Control in LLMs via Steering Target Atoms](https://mlsanity.com/paper/2505.20322/) (2025)
- [The Geometry of Reasoning: Flowing Logics in Representation Space](https://mlsanity.com/paper/2510.09782/) (2025)
- [Large Language Model Reasoning Failures](https://mlsanity.com/paper/2602.06176/) (2026)
- [Rethinking Evaluation of Sparse Autoencoders through the Representation of Polysemous Words](https://mlsanity.com/paper/2501.06254/) (2025)
- [Tracking the Feature Dynamics in LLM Training: A Mechanistic Study](https://mlsanity.com/paper/2412.17626/) (2024)
- [Do Sparse Autoencoders Identify Reasoning Features in Language Models?](https://mlsanity.com/paper/2601.05679/) (2026)
- [From Data Statistics to Feature Geometry: How Correlations Shape Superposition](https://mlsanity.com/paper/2603.09972/) (2026)
- [Understanding Emergent Misalignment via Feature Superposition Geometry](https://mlsanity.com/paper/2605.00842/) (2026)
- [Sparse Feature Coactivation Reveals Causal Semantic Modules in Large Language Models](https://mlsanity.com/paper/2506.18141/) (2025)
- [CSRv2: Unlocking Ultra-Sparse Embeddings](https://mlsanity.com/paper/2602.05735/) (2026)
- [From Atoms to Trees: Building a Structured Feature Forest with Hierarchical Sparse Autoencoders](https://mlsanity.com/paper/2602.11881/) (2026)
- [Analyzing Hierarchical Structure in Vision Models with Sparse Autoencoders](https://mlsanity.com/paper/2505.15970/) (2025)
- [The Shape of Adversarial Influence: Characterizing LLM Latent Spaces with Persistent Homology](https://mlsanity.com/paper/2505.20435/) (2025)
- [Harmonic Loss Trains Interpretable AI Models](https://mlsanity.com/paper/2502.01628/) (2025)
- [On the creation of narrow AI: hierarchy and nonlocality of neural network skills](https://mlsanity.com/paper/2505.15811/) (2025)
- [Embryology of a Language Model](https://mlsanity.com/paper/2508.00331/) (2025)
- [The Compositional Architecture of Regret in Large Language Models](https://mlsanity.com/paper/2506.15617/) (2025)
- [Latent Concept Disentanglement in Transformer-based Language Models](https://mlsanity.com/paper/2506.16975/) (2025)
- [Representation Learning on a Random Lattice](https://mlsanity.com/paper/2504.20197/) (2025)
- [Sparse Autoencoders Map Brain-LLM Alignment onto Cortical Semantic Topography](https://mlsanity.com/paper/2605.23035/) (2026)
- [The Logical Implication Steering Method for Conditional Interventions on Transformer Generation](https://mlsanity.com/paper/2502.03618/) (2025)
- [What's in a prompt? Language models encode literary style in prompt embeddings](https://mlsanity.com/paper/2505.17071/) (2025)
- [Fusion Steering: Prompt-Specific Activation Control](https://mlsanity.com/paper/2505.22572/) (2025)
- [Graph-Regularized Sparse Autoencoders for LLM Safety Steering](https://mlsanity.com/paper/2512.06655/) (2025)
- [How Do Large Language Models Learn Concepts During Continual Pre-Training?](https://mlsanity.com/paper/2601.03570/) (2026)

## Abstract (arXiv)

Sparse autoencoders have recently produced dictionaries of high-dimensional vectors corresponding to the universe of concepts represented by large language models. We find that this concept universe has interesting structure at three levels: 1) The "atomic" small-scale structure contains "crystals" whose faces are parallelograms or trapezoids, generalizing well-known examples such as (man-woman-king-queen). We find that the quality of such parallelograms and associated function vectors improves greatly when projecting out global distractor directions such as word length, which is efficiently done with linear discriminant analysis. 2) The "brain" intermediate-scale structure has significant spatial modularity; for example, math and code features form a "lobe" akin to functional lobes seen in neural fMRI images. We quantify the spatial locality of these lobes with multiple metrics and find that clusters of co-occurring features, at coarse enough scale, also cluster together spatially far more than one would expect if feature geometry were random. 3) The "galaxy" scale large-scale structure of the feature point cloud is not isotropic, but instead has a power law of eigenvalues with steepest slope in middle layers. We also quantify how the clustering entropy depends on the layer.