The Linear Representation Hypothesis Needs a Group Action
cs.LG, cs.AI, cs.CL
Submitted: 2026-09-22
Updated: 2026-09-22
Comments: 16 pages, 1 table
License: http://creativecommons.org/licenses/by/4.0/
The gist: To make claims about representations that generalize beyond a particular trained model, we need to specify when two representations should count as equivalent.
Terminology
Abstract
To make claims about representations that generalize beyond a particular trained model, we need to specify when two representations should count as equivalent. The Linear Representation Hypothesis is often discussed without making this equivalence explicit. Different notions of equivalence preserve different structures, so metrics, probes, and interventions that appear to study the same representation may in fact correspond to different hypotheses. We therefore argue that the Linear Representation Hypothesis is not one hypothesis but a family of claims distinguished by representation equivalence. We formalize this idea using group actions, specifying the representation object, the procedure that produces it, and the property ultimately asserted, while accounting for equivalences imposed by the model architecture. This framework clarifies how assumptions can change across metrics, reading points, and analysis stages, and we use it to audit common representation quantities and recent interpretability analyses.
Sources
- Understanding intermediate layers using linear classifier probes
- Cross-Lingual Steering for Figurative Language Generation
- Curveball Steering: The Right Direction To Steer Isn't Always Linear
- How Many Features Can a Language Model Store Under the Linear Representation Hypothesis?
- Toy Models of Superposition
- Towards Unifying Interpretability and Control: Evaluation via Intervention
- Steering Language Models With Activation Engineering
- Multi-Head Attention as Ensemble Nadaraya-Watson Estimation: Variance Reduction, Decorrelation, and Optimal Head Diversity
- PRISM-$\Delta$: Differential Subspace Steering for Prompt Highlighting in Large Language Models
- Measuring Affinity between Attention-Head Weight Subspaces via the Projection Kernel
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