Manifold Steering Reveals the Shared Geometry of Neural Network Representation and Behavior

TL;DR

This paper demonstrates that understanding and manipulating the geometric structure of neural representations enables more natural and effective control of model behavior across various tasks and modalities.

Contribution

It introduces manifold steering, a method that respects activation space geometry, revealing a bidirectional link between neural representation geometry and behavior.

Findings

Steering along the activation manifold yields natural behavioral trajectories.

Linear steering in Euclidean space produces off-manifold, unnatural outputs.

Optimizing interventions along behavior manifolds traces the curvature of activation manifolds.

Abstract

Neural representations carry rich geometric structure; but does that structure causally shape behavior? To address this question, we intervene along paths through activation space defined by different geometries, and measure the behavioral trajectories they induce. In particular, we test whether interventions that respect the geometry of activation space will yield behaviors close to those the model exhibits naturally. Concretely, we first fit an activation manifold $M_h$ to representations and a behavior manifold $M_y$ to output probability distributions. We then test the link $M_h \leftrightarrow M_y$ via interventions: we find that steering along $M_h$, which we term manifold steering, yields behavioral trajectories that follow $M_y$, while linear steering -- which assumes a Euclidean geometry -- cuts through off-manifold regions and hence produces unnatural outputs. Moreover, optimizing interventions in activation space to produce paths along $M_y$ recovers activation trajectories that trace the curvature of $M_h$. We demonstrate this bidirectional relationship between the geometry of representation and behavior across tasks and modalities. In language models, we use reasoning tasks with cyclic and sequential geometries as well as in-context learning tasks with more complex graph geometries. In a video world model, we use a task with geometry corresponding to physical dynamics. Overall, our work shows that geometry in neural representation is not merely incidental, but is in fact the proper object for enabling principled control via intervention on internals. This recasts the core problem of steering from finding the right direction to finding the right geometry.