A Geometric Account of Activation Steering through Angle-Norm Decomposition

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 steering provides a lightweight way to control large language models (LLMs) by modifying their hidden activations at inference time. Among these approaches, norm-preserving steering aims to change model behavior without altering the activation norm, reducing the risk of representation collapse and degradation. However, existing norm-preserving methods are limited by predefined steering trajectories and by their reliance on one-step updates, which may fail to capture the complex structure of activation distributions. We propose GeoSteer, an optimization-based method for norm-preserving activation steering. GeoSteer formulates steering as a Riemannian optimization problem and updates activations through a sequence of small geodesic steps on the representation manifold. To avoid fixed steering directions, GeoSteer learns a nonlinear activation-space objective that distinguishes desired from undesired activations, and uses this function to adaptively guide each steering step. This multistep formulation yields smoother, more stable, and more consistent steering behavior while preserving the activation norm. Across TruthfulQA, RealToxicityPrompts, and UltraFeedback benchmarks, GeoSteer consistently improves over state-of-the-art activation steering baselines. These results suggest that norm-preserving steering can be made more effective by replacing predefined one-step edits with adaptive, geometry-aware optimization.

Activation steering has emerged as a promising alternative for controlling language-model behavior at inference time by modifying intermediate representations while keeping model parameters frozen. However, large-scale evaluations such as AxBench show that existing steering methods are often outperformed by simple in-context prompting and generalize poorly to unseen concepts. We hypothesize that these limitations arise from unvalidated simplifying assumptions shared across prior methods, which typically restrict steering interventions to fixed, single-step, position-invariant transforms. We propose FLAS (Flow-based Activation Steering), which learns a general, concept-conditioned velocity field $v_t(h,t,c)$ that transports unsteered activations to steered ones without relying on these assumptions. On AxBench, FLAS is the first learned method to consistently outperform prompting, reaching held-out harmonic means of $1.015$ on Gemma-2-2B-IT and $1.113$ on Gemma-2-9B-IT without per-concept tuning. Analysis of the learned flow shows curved, multi-step, token-varying trajectories, which suggests that previous hypotheses on activation space geometry might be incomplete.