A Mathematical Theory of the Jacobian Lens

Recent research introduces a mathematical theory for the Jacobian lens, an interpretability tool developed by Anthropic. This method identifies linear directions within the hidden states of frontier language models, which resemble working memory-like representations. The theory demonstrates that the J-lens preferentially detects coherent features, specifically directions acting as leading, spatially coherent Lyapunov modes. These specific directions are characterized by the fact that small perturbations along them amplify across layers, clarifying the mechanistic properties of these features.