The Information Geometry of Softmax: Probing and Steering

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.

Steering is a widely used technique for controlling large language models, yet its effects are often unstable and hard to predict. Existing theoretical accounts are largely based on the Linear Representation Hypothesis (LRH). While LRH assumes that concepts can be orthogonalized for lossless control, this idealized mapping fails in real representations and cannot account for the observed unpredictability of steering. By relaxing LRH's orthogonality assumption while preserving linear representations, we show that overlapping concept contributions naturally yield a sample-specific axis-orthogonal structure. We formalize this as the Cylindrical Representation Hypothesis (CRH). In CRH, a central axis captures the main difference between concept absence and presence and drives concept generation. A surrounding normal plane controls steering sensitivity by determining how easily the axis can activate the target concept. Within this plane, only specific sensitive sectors strongly facilitate concept activation, while other sectors can suppress or delay it. While the surrounding normal plane can be reliably identified from difference vectors, the sensitive sector cannot, introducing intrinsic uncertainty at the sector level. This uncertainty provides a principled explanation for why steering outcomes often fluctuate even when using well-aligned directions. Our experiments verify the existence of the cylindrical structure and demonstrate that CRH provides a valid and practical way to interpret model steering behavior in real settings: https://github.com/mbzuai-nlp/CRH.

Activation-based steering provides control of LLM behavior at inference time, but the dominant paradigm reduces each concept to a single direction whose geometry is left largely unexamined. Rather than selecting a single steering direction, we use conceptors: soft projection matrices estimated from activations pooled across both poles of a bipolar concept, which preserve the concept's full multidimensional subspace. A geometric analysis shows the bipolar subspace strictly subsumes the single-vector baseline. We further show that the conceptor quota provides a parameter-free layer-selection diagnostic, predicting concept separability with Pearson correlations up to r=0.96 across three instruction-tuned models and three semantic dimensions. Beyond selection, conceptors admit a closed-form Boolean algebra (AND, OR, NOT): we evaluate conceptor compositionality on thematically related sub-concepts. Across a systematic five-axis design-space evaluation, conceptors match or outperform additive baselines at layers where concept subspaces are multi-dimensional while producing substantially fewer degenerate outputs. Conceptor steering is a geometrically principled, compositional, and practically safer alternative to single-direction steering from a limited number of contrastive pairs.