From Directions to Regions: Decomposing Activations in Language Models via Local Geometry

Linear activation steering has gained popularity as a simple and empirically effective way to control language model behavior. More recently, spherical steering paradigms have been proposed to address limitations of additive interventions, often motivated by the assumption that hidden-state norm does not carry concept-relevant information. In this work, we revisit this assumption through a controlled empirical study designed to disentangle the roles of angular and radial components. We show that steering methods differ mainly in how they couple two geometric effects: changing a token's angular alignment with a concept direction and changing its hidden-state norm. Across seven language models, we find that concepts are represented primarily in angular structure, supporting the motivation for spherical methods, but that norm remains important for the stability and downstream effects of steering. Our results explain why interventions with similar concept-level effects can behave differently, and suggest that activation steering should be parameterized by interpretable angular and radial components of the intervention, rather than by a single additive coefficient that entangles these two effects.

Fine-grained control of language model behaviors (e.g., steering) is among the more actionable outcomes of interpretability research. For binary concepts such as refusal, a single direction in activation space often suffices for steering. However, many concepts are not binary: Animals and Countries contain many subcategories, each with multiple instances. For these concepts, the search space over possible representation geometries is far larger than for binary concepts; it is thus not clear what geometries are most appropriate, nor what methods are most effective at recovering them. In this work, we introduce MAxBench, a geometry-agnostic evaluation framework for multinomial concept representations based on sampling from the recovered concept representation. We use MAxBench to compare 10 localization methods (covering 5 geometry types) across 6 concepts and 4 models. Using this framework, we find that (i) affine subspaces steer more reliably and have greater recall than rank-one or linear subspaces; (ii) much of this advantage is due to better non-zero offsets rather than the choice of bases; (iii) manifold steering is competitive with the best methods when applicable; and (iv) no method consistently outperforms prompting, in alignment with prior findings on binary concepts. These findings underscore the importance of expanding the scope of interpretability research and meta-evaluation to concepts with more varied structure.

Activation steering has emerged as a key methodology for controlling the behavior of large language models (LLMs). Existing difference-in-means based methods, however, are fundamentally limited: they capture only mean differences between class activations and fail to recover discriminative signals that naturally exist in the nonlinear feature subspace under the superposition hypothesis. Motivated by that, we propose High-Dimensional Random-projection for Activation Steering (HiDRA), a training-free approach that integrates seamlessly with existing activation steering methods. By performing activation addition in the projected high-dimensional space, HiDRA can provably capture a better discriminative structure beyond the reach of linear methods. Experiments across diverse LLM families and benchmarks demonstrate that HiDRA consistently outperforms baseline counterparts, achieving stronger behavioral control without significant computational overhead.