Why the Jacobian lens favors slow features: a theory writes its choice as three coherence factors and predicts their ordering in toy models
The Jacobian lens (J-lens), an interpretability tool that decodes a model's hidden states into words, had been used to find working-memory-like linear directions, but what makes a feature detectable to it was unclear. This work gives a mathematical theory: the lens preferentially selects directions that grow across layers, stay smooth across token positions, and remain consistent across contexts, that is, coherent Lyapunov modes. On a Bayes filter and a two-layer toy transformer, changing task parameters to modulate coherence shifts the ordering of latent states the lens selects, as predicted.