Linear representation understanding requires specifying equivalence clearly
The Linear Representation Hypothesis Needs a Group Action
Machine LearningArtificial IntelligenceComputation and Language
Summary
When studying how AI systems understand information, it’s important to say when two ways of representing that information are considered the same. The authors found that people often talk about these representations without being clear about what ‘equivalent’ means. They explain that different definitions of equivalence can lead to very different conclusions about how models work. To fix this, they use a mathematical idea called group actions to clearly define and analyze these equivalences, which helps clarify how we interpret AI representations.
What this means in practice
- •For machine learning engineers: Design interpretability tests that correctly account for when two learned representations should be considered equivalent, improving model analysis reliability.
- •For ai tool developers: Build diagnostic tools that explicitly handle representation equivalence, avoiding misleading conclusions about model internals during analysis.
A position paper. It proposes an approach and reports no results.
Authors
Louie Hong Yao, Yuhao Li, Shengchao Liu
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.