The Linear Representation Hypothesis Needs a Group Action

The paper argues that the Linear Representation Hypothesis (LRH) should not be treated as a single claim but as a family of claims differentiated by how representations are considered equivalent. It highlights that different equivalence notions preserve different structures, leading to metrics, probes, and interventions that may actually test distinct hypotheses. By formalizing these ideas with group actions, the authors provide a framework that clarifies how assumptions vary across metrics, reading points, and analysis stages, and they apply it to audit common representation quantities and recent interpretability analyses.

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