# The Linear Centroids Hypothesis: Features as Directions Learned by Local Experts

arXiv: https://arxiv.org/abs/2604.11962 · HTML: https://mlsanity.com/paper/2604.11962/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 41 of its 55 references that are in the map.
- First posted on arXiv 13 Apr 2026.

## Builds on (most cited first, 30 of 41)

- [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)
- [Swin Transformer: Hierarchical Vision Transformer using Shifted Windows](https://mlsanity.com/paper/2103.14030/) (2021)
- [DINOv2: Learning Robust Visual Features without Supervision](https://mlsanity.com/paper/2304.07193/) (2023)
- [Grad-CAM: Visual Explanations from Deep Networks via Gradient-based Localization](https://mlsanity.com/paper/1610.02391/) (2016)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [A ConvNet for the 2020s](https://mlsanity.com/paper/2201.03545/) (2022)
- [Gaussian Error Linear Units (GELUs)](https://mlsanity.com/paper/1606.08415/) (2016)
- [Axiomatic Attribution for Deep Networks](https://mlsanity.com/paper/1703.01365/) (2017)
- [Locating and Editing Factual Associations in GPT](https://mlsanity.com/paper/2202.05262/) (2022)
- [Deep Inside Convolutional Networks: Visualising Image Classification Models and Saliency Maps](https://mlsanity.com/paper/1312.6034/) (2013)
- [Describing Textures in the Wild](https://mlsanity.com/paper/1311.3618/) (2013)
- [BLOOM: A 176B-Parameter Open-Access Multilingual Language Model](https://mlsanity.com/paper/2211.05100/) (2022)
- [Sparse Autoencoders Find Highly Interpretable Features in Language Models](https://mlsanity.com/paper/2309.08600/) (2023)
- [Interpretability Beyond Feature Attribution: Quantitative Testing with Concept Activation Vectors (TCAV)](https://mlsanity.com/paper/1711.11279/) (2017)
- [Interpretability in the Wild: a Circuit for Indirect Object Identification in GPT-2 small](https://mlsanity.com/paper/2211.00593/) (2022)
- [Sanity Checks for Saliency Maps](https://mlsanity.com/paper/1810.03292/) (2018)
- [SmoothGrad: removing noise by adding noise](https://mlsanity.com/paper/1706.03825/) (2017)
- [Toy Models of Superposition](https://mlsanity.com/paper/2209.10652/) (2022)
- [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)
- [RobustBench: a standardized adversarial robustness benchmark](https://mlsanity.com/paper/2010.09670/) (2020)
- [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)
- [Emergent Linear Representations in World Models of Self-Supervised Sequence Models](https://mlsanity.com/paper/2309.00941/) (2023)
- [Localizing Model Behavior with Path Patching](https://mlsanity.com/paper/2304.05969/) (2023)
- [fastai: A Layered API for Deep Learning](https://mlsanity.com/paper/2002.04688/) (2020)
- [Open Problems in Mechanistic Interpretability](https://mlsanity.com/paper/2501.16496/) (2025)
- [Complexity of Linear Regions in Deep Networks](https://mlsanity.com/paper/1901.09021/) (2019)
- [A Comprehensive Study on Robustness of Image Classification Models: Benchmarking and Rethinking](https://mlsanity.com/paper/2302.14301/) (2023)

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

- [Gradient-Direction Sensitivity Reveals Linear-Centroid Coupling Hidden by Optimizer Trajectories](https://mlsanity.com/paper/2604.25143/) (2026)
- [When Do Attention Circuits Form? Developmental Trajectories of Capability and Attention-Sink Emergence Across Three 1B-ClassArchitectures](https://mlsanity.com/paper/2606.02378/) (2026)
- [Behavioral Monitoring of JEPA World Models with Jacobian Centroids](https://mlsanity.com/paper/2609.33940/) (2026)

## Abstract (arXiv)

The Linear Representation Hypothesis (LRH) identifies features of a trained deep network (DN) as linear directions in the activation spaces, i.e., output spaces of intermediate layers. This characterization decouples the input-output maps learned by a DN from the organization of feature directions in its activation spaces. We introduce the Linear Centroids Hypothesis (LCH), which instead identifies features with linear directions among a DN's centroid spaces -- where any vector denotes a centroid or summary of a local affine expert characterizing the learned input-output maps of the DN exactly (e.g., for piecewise-affine DNs) or approximately (e.g., for smooth DNs like transformers). We show that replacing intermediate activations with centroids yields a functional drop-in alternative for standard interpretability tools. Empirically, this change yields sparser, more downstream-useful feature dictionaries on DINO ViTs, suppresses spurious directions on a controlled task, recovers interpretable circuits in GPT2-Large, and produces faithful gradient-based saliency maps. LCH unifies dictionaries, probing, circuits, and saliency maps into a single geometric object grounded in the network's input-output map -- making interpretability mechanistic by construction rather than post hoc. Code to study the LCH https://github.com/ThomasWalker1/LinearCentroidsHypothesis .