# A Geometric View for Understanding Concept Learning and Neuron Interpretation in Sparse Autoencoders

arXiv: https://arxiv.org/abs/2606.07007 · HTML: https://mlsanity.com/paper/2606.07007/

## Key facts

- Cited by 1 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 32 of its 49 references that are in the map.
- First posted on arXiv 5 Jun 2026.

## Placed in

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

## Builds on (most cited first, 30 of 32)

- [Sparse Autoencoders Find Highly Interpretable Features in Language Models](https://mlsanity.com/paper/2309.08600/) (2023)
- [Concept Bottleneck Models](https://mlsanity.com/paper/2007.04612/) (2020)
- [Toy Models of Superposition](https://mlsanity.com/paper/2209.10652/) (2022)
- [In-context Learning and Induction Heads](https://mlsanity.com/paper/2209.11895/) (2022)
- [The Linear Representation Hypothesis and the Geometry of Large Language Models](https://mlsanity.com/paper/2311.03658/) (2023)
- [Scaling and evaluating sparse autoencoders](https://mlsanity.com/paper/2406.04093/) (2024)
- [On the Number of Linear Regions of Deep Neural Networks](https://mlsanity.com/paper/1402.1869/) (2014)
- [Finding Neurons in a Haystack: Case Studies with Sparse Probing](https://mlsanity.com/paper/2305.01610/) (2023)
- [Jumping Ahead: Improving Reconstruction Fidelity with JumpReLU Sparse Autoencoders](https://mlsanity.com/paper/2407.14435/) (2024)
- [Foundational Challenges in Assuring Alignment and Safety of Large Language Models](https://mlsanity.com/paper/2404.09932/) (2024)
- [Scaling Monosemanticity: Extracting Interpretable Features from Claude 3 Sonnet](https://mlsanity.com/paper/2605.29358/) (2026)
- [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)
- [On the number of response regions of deep feed forward networks with piece-wise linear activations](https://mlsanity.com/paper/1312.6098/) (2013)
- [Sparse Autoencoders Do Not Find Canonical Units of Analysis](https://mlsanity.com/paper/2502.04878/) (2025)
- [InterPLM: Discovering Interpretable Features in Protein Language Models via Sparse Autoencoders](https://mlsanity.com/paper/2412.12101/) (2024)
- [Explaining black box text modules in natural language with language models](https://mlsanity.com/paper/2305.09863/) (2023)
- [Projecting Assumptions: The Duality Between Sparse Autoencoders and Concept Geometry](https://mlsanity.com/paper/2503.01822/) (2025)
- [From Flat to Hierarchical: Extracting Sparse Representations with Matching Pursuit](https://mlsanity.com/paper/2506.03093/) (2025)
- [Language Models are Injective and Hence Invertible](https://mlsanity.com/paper/2510.15511/) (2025)
- [Do Sparse Autoencoders Capture Concept Manifolds?](https://mlsanity.com/paper/2604.28119/) (2026)
- [Disentangling Dense Embeddings with Sparse Autoencoders](https://mlsanity.com/paper/2408.00657/) (2024)
- [Forward-Backward Selection with Early Dropping](https://mlsanity.com/paper/1705.10770/) (2017)
- [Because we have LLMs, we Can and Should Pursue Agentic Interpretability](https://mlsanity.com/paper/2506.12152/) (2025)
- [The Shape of Beliefs: Geometry, Dynamics, and Interventions along Representation Manifolds of Language Models' Posteriors](https://mlsanity.com/paper/2602.02315/) (2026)
- [Incorporating Hierarchical Semantics in Sparse Autoencoder Architectures](https://mlsanity.com/paper/2506.01197/) (2025)
- [From Directions to Regions: Decomposing Activations in Language Models via Local Geometry](https://mlsanity.com/paper/2602.02464/) (2026)
- [Understanding sparse autoencoder scaling in the presence of feature manifolds](https://mlsanity.com/paper/2509.02565/) (2025)
- [A Mathematical Philosophy of Explanations in Mechanistic Interpretability -- The Strange Science Part I.i](https://mlsanity.com/paper/2505.00808/) (2025)
- [Evaluating Neuron Explanations: A Unified Framework with Sanity Checks](https://mlsanity.com/paper/2506.05774/) (2025)

## Built on by (most cited first, 1 of 1)

- [A Dominant Diffuse Phase in the Sparse Autoencoder Phase Diagram](https://mlsanity.com/paper/2609.10299/) (2026)

## Abstract (arXiv)

We propose a unified mathematical framework for a geometric understanding of concept learning and neuron interpretation in sparse autoencoders (SAEs). While SAEs improve interpretability of neural networks by learning sparse feature representations, a principled definition of ''concept'' and ''learning'' remains unclear. We formalize concepts as sets of data points and cast concept learning as a set-alignment problem between human-defined and model-induced concepts. This formulation distinguishes three increasingly strong notions of learning -- detection, separation, and approximation -- and yields geometric conditions, error bounds, and capacity constraints for when concepts can be represented by individual neurons or multi-neuron units. It also provides a set-theoretic account for common SAE phenomena, including feature splitting, feature absorption, feature families, and hierarchical concepts. Finally, we connect concept learning and neuron interpretation through formal concept analysis, showing that the two directions need not agree and that their many-to-many structure can be organized by concept lattices. Experiments on synthetic data with ReLU and Top-K SAEs illustrate the theory and reveal the effects of SAE size and sparsity on concept learning.