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A Geometric Account of Activation Steering through Angle-Norm Decomposition

Georgii Aparin, Tatiana Gaintseva

arXiv:2606.06735v2cs.AI

TL;DR

The study asks whether concept information is primarily angular and whether preserving hidden-state norm is always the right steering constraint. It disentangles angular and radial intervention effects and finds that angle governs semantic control while radius affects steering stability.

  • Problem

    Existing steering methods entangle angular movement toward a concept direction with hidden-state norm changes, leaving the roles of direction and magnitude insufficiently examined.

  • Method

    The study parameterizes steering by separately controlling angular movement toward a concept direction and radial changes in hidden-state norm.

  • Results

    Across seven language models and four concept datasets, concepts are represented primarily in activation direction, while norm changes affect stability and capability preservation.

  • Takeaways & Limitations

    Activation steering is best understood as a two-parameter intervention in which angle controls the concept and radius controls intervention stability.

  • Takeaways & Limitations

    The experiments cover a limited set of models and concepts, so other architectures or more complex behaviors may exhibit different geometry.

Abstract

from arXiv · show

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.

1 Introduction

The paper reframes activation steering as a two-parameter geometric intervention: angle primarily controls concept information, while radius affects stability and capability preservation. Across seven language models and four concept datasets, the authors find that concepts are encoded mainly in activation direction, but modest norm increases can improve strong steering.

  • Geometric framework: The study decomposes hidden states into angular alignment with a concept direction and radial norm, comparing six steering methods by their angular and norm constraints.Methods vary in whether they preserve norm and enforce a per-token target concept score.
  • Angular encoding: Across seven language models and four concept datasets, normalized-state probes closely match raw-state probes, whereas norm-only probes remain near chance.These results support the hypothesis that concept-discriminative information is primarily encoded in activation direction rather than magnitude.
  • Radial effects: Norm remains important for generation stability and capability preservation: strict norm preservation at high angular strength can increase perplexity and degrade capabilities.Methods reaching the same angular target with a modest norm increase often better preserve fluency and downstream performance.
  • Design implications: The authors therefore propose parameterizing steering by interpretable angle and radius, since methods with similar concept-level effects can differ in stability, input relevance, and capability preservation.Angle controls the intended semantic effect, while radius influences generation stability, input relevance, and capability preservation.
  • Radial effects: At high steering strengths, modest norm increases can reduce perplexity by up to 1.8× without substantially changing the semantic steering effect.This result is illustrated by the norm-scaling analysis and supports treating radius as an independent steering parameter.

2 Related Work

Prior work uses activation directions to control behavior and has introduced spherical alternatives motivated by angular concept representations and norm preservation. This paper separates angular and radial effects, finding that direction primarily encodes concepts while norm influences stability and capability preservation.

  • Activation steering and representation engineering: Activation steering modifies intermediate activations without updating weights, using behavior-associated hidden-state directions to affect attributes including truthfulness, sentiment, topic, refusal, and toxicity.Examples include ITI, ActAdd, CAA, and Representation Engineering.
  • Linear concept representations: Linear concept representations motivate contrastive direction extraction, probing, and direction-based interventions, but additive steering simultaneously changes angular alignment and representation norm.Identifying a useful concept direction does not determine how intervention geometry should be controlled.
  • Angular and spherical steering: Angular Steering rotates activations, whereas Spherical Steering applies a norm-preserving geodesic rotation toward a target direction to preserve activation norm.These methods are motivated by the idea that concept information is primarily angular and norm preservation helps maintain generation quality.
  • Adaptive and token-wise steering: Adaptive steering methods vary intervention sites or strengths across heads, layers, tokens, or examples because a single global coefficient may be insufficient.The paper frames the relevant adaptation as controlling angular and radial components separately.
  • Our contribution: The evaluated concepts are primarily encoded in activation direction, while modest radial changes can substantially affect perplexity and capability preservation at fixed angular concept score.The paper therefore reframes steering as a two-parameter intervention over angle and radius rather than a one-dimensional additive choice.

3 Methodology

The methodology treats activation steering as a geometric intervention that separates angular movement toward a concept direction from changes in hidden-state norm. It constructs a shared contrastive concept direction and compares steering operations under controlled angular and radial constraints.

  • Geometric decomposition: Hidden states are decomposed into a norm, unit direction, angular concept score, and orthogonal residual direction, separating angular and radial steering effects.The target concept score γ specifies the desired angular alignment within the subspace span(s, v).
  • Concept direction: Each model-dataset-fold cell uses the same unit-normalized contrastive mean-difference direction, isolating intervention geometry from direction-estimation differences.The direction is estimated from N = 256 positive-negative completion pairs using residual-stream activations at the last prompt token.
  • Steering methods: The six steering operations vary in whether they preserve hidden-state norm and whether they target each token’s concept score independently.CAA is fixed-strength and neither norm-preserving nor per-token calibrated, whereas CAA-r preserves the original norm without enforcing a target score.
  • Steering methods: CAA-m and Spherical Steering share the same angular component, while differing only in their radial component; renormalized CAA-m is exactly equivalent to Spherical Steering.CAA-m reaches target score γ while allowing norm changes, whereas Spherical Steering preserves norm and enforces the target score per token.
  • Steering methods: SN adds a fixed multiplicative norm factor β to Spherical Steering without changing its angular component, with β = 1 reducing exactly to S.This parameter tests whether norm functions as a stability parameter after semantic angular control is fixed.
  • Experimental design: Four controlled experiments isolate angular control and norm modification, beginning by measuring hidden-state norm variation across layers and token populations.The first experiment computes coefficient of variation for last-prompt-token, all-prompt-token, and generated-token norms across multiple corpora.

2. Angular versus radial concept encoding.

This section tests whether concepts are encoded primarily in hidden-state direction or magnitude. It then uses matched angular control and isolated norm scaling to separate the effects of angle and radius on steering.

  • Angular versus radial concept encoding: Three linear probes compare raw states, normalized states, and scalar norms to test whether concept information is angular or radial.Matching raw and normalized probes with near-chance norm-only probes would indicate primarily angular encoding.
  • Angular versus radial concept encoding: Steering methods are compared at matched per-token or mean concept scores across task performance, concept-score variance, norm ratio, perplexity, and capability metrics.Per-token methods target γ, while fixed-strength methods use binary search to match the desired mean score ¯γ.
  • Angular versus radial concept encoding: Holding γ fixed while varying β isolates whether radius changes improve generation stability without changing angular semantic control.β changes only the norm and leaves the angular concept score fixed.
  • Angular versus radial concept encoding: Held-out direction splits, separate evaluation examples, and parameter calibration precede downstream measurements for each steering method.Calibration searches α for CAA and CAA-r, ∆θ for AS, and token-specific α for CAA-m.
  • Angular versus radial concept encoding: The experiments distinguish whether steering success arises from angular movement, strict norm preservation, or interaction between angle and radius.Methods are compared after matching either per-token concept scores or mean concept scores.

4 Experiments

Experiments across seven language models show that concept information is encoded primarily in angular structure, while hidden-state norm remains a separate factor affecting steering stability and generation quality. Matched comparisons indicate that steering should be parameterized by distinct angular and radial components rather than a single entangled strength coefficient.

  • Hidden-state geometry: Norm concentration is architecture-dependent, with Llama and Qwen generally more concentrated than Gemma at middle and later layers.This motivates treating norm as a separate geometric degree of freedom even when semantic information is primarily angular.
  • Concept representation: Normalized probes nearly match raw hidden-state probes across models and datasets, while norm-only probes remain near chance, indicating primarily angular concept encoding.The evaluated concepts are therefore represented mainly in direction rather than magnitude.
  • Angular–radial comparisons: At matched per-token targets, S and CAA-m share angular control but differ radially, and CAA-m is generally more stable than strict norm-preserving S at high γ.S can incur large perplexity penalties and capability loss, whereas CAA-m often retains lower perplexity and higher MMLU accuracy.
  • Fixed-strength steering: CAA and CAA-r have similar downstream behavior, whereas norm-preserving AS differs substantially, showing that per-token concept-score distributions affect steering quality beyond norm preservation.CAA-r only rescales the additive result, while AS applies a fixed spherical displacement distributed differently across tokens.
  • Angular–radial parameterization: 1.8× perplexity improvement at γ = 0.7 results from moving β = 1.0 to β = 1.2, while task metrics remain within about 2.5 percentage points.These results support a two-parameter view: angular control determines semantic effect, while radial scale influences coherent generation and stability.

5 Conclusion

The conclusion frames activation steering as a two-parameter intervention that separates angular movement toward a concept direction from radial changes in hidden-state norm. Across seven language models and four concept datasets, concepts are represented primarily in activation direction, while radius affects stability, perplexity, and capability preservation.

  • Geometric decomposition: Additive interventions entangle angular movement toward a concept direction with radial changes in hidden-state norm, making one coefficient difficult to interpret across tokens.The same additive coefficient can induce different angular shifts and norm changes depending on each token’s initial geometry.
  • Concept representation: Across seven language models and four concept datasets, concepts are represented primarily in activation direction.Normalized probes closely match raw probes, while norm-only probes remain near chance, supporting largely angular semantic control.
  • Radial effects: Radial changes can strongly affect perplexity and capability preservation even when angular concept scores remain fixed.Strict norm preservation can become unstable at high steering strengths, whereas modest norm increases reduce degradation without materially changing semantic effects.
  • Two-parameter steering: Activation steering is best reframed as a two-parameter intervention in which angle controls the intended concept and radius controls intervention stability.This framework explains why methods with similar concept-level effects can behave differently and provides a more interpretable basis for future token-wise steering methods.

6 Limitations

The study’s conclusions are limited by its fixed-layer evaluation, narrow model and concept coverage, shared steering-direction construction, and discrete norm-scaling experiments. These choices constrain how broadly the angle–norm findings and norm-selection guidance can be generalized.

  • Layer selection: Steering is evaluated only at a fixed layer chosen at 75% depth, although the optimal angle–norm trade-off may vary across layers.The fixed layer enables controlled method comparisons but may not identify the best trade-off elsewhere.
  • Models and concepts: Experiments cover only Llama, Qwen, and Gemma models and truthfulness, sentiment, and toxicity-related steering, so other architectures or complex behaviors may differ geometrically.The limited evaluation scope leaves open whether the observed geometry generalizes beyond these models and concepts.
  • Steering directions: All methods use the same contrastive mean-difference steering direction, isolating intervention geometry without testing alternative direction-estimation methods.Thus, the conclusions are not established for steering directions constructed differently.
  • Norm scaling: Norm-scaling experiments use a small discrete set of β values and do not provide an automatic rule for selecting the best norm scale for a new model, layer, or task.The experiments identify norm as an important stability parameter but leave norm-scale selection unresolved.

A Additional Norm-Variation Analysis

The appendix shows that norm variation is strongly position- and architecture-dependent for prompt tokens but comparatively stable for generated tokens. These findings support preserving each token’s own radius during spherical steering rather than imposing a shared global radius.

  • Per-corpus norm variation: Llama and Qwen models usually show moderate last-prompt-token norm CV, whereas Gemma shows substantially larger variation because of its post-norm architecture.
  • Prompt-token positions: The first prompt token has a distinct, architecture-dependent norm distribution, after which CV settles into a more stable plateau.In Llama and Qwen models, the first token behaves like an attention-sink position; Gemma remains elevated across many layers.
  • Generation-token positions: Generated-token positions are more stable than prompt positions for most instruction-tuned models, while the Llama base model shows larger CV at later layers.This is the regime most relevant to the steering hook during decoding.
  • Token-population and cumulative variation: Pooling all prompt tokens produces much larger CV because positions have different typical norm scales, whereas generation-token CV curves converge quickly.Mean norm profiles likewise show strong prompt-position effects but nearly constant generation-token norms across positions at a fixed layer for most instruction-tuned models.
  • Implication for spherical steering: Norm preservation in spherical steering should preserve each token’s own radius rather than force all activations onto a shared global radius.Prompt-token norms are strongly position-dependent, while generation-token norms are more stable across decoding steps.

B Additional Directional-Encoding Results · C CAA-m Per-Token Matching Algorithm

Additional probe results show that concept information is primarily directionally encoded, while norm-only features remain near chance even when radial variability is large. CAA-m separately selects each token’s additive coefficient to match a requested normalized concept score, controlling angle while allowing norm changes.

  • B Additional Directional-Encoding Results: Unit-normalized probes closely match raw-hidden-state probes across evaluated concepts, showing that removing the radial component causes almost no loss in probe accuracy.The comparison covers raw hidden states, unit-normalized hidden states, and norm-only features.
  • B Additional Directional-Encoding Results: Norm-only probes remain near chance across datasets and model families, indicating that activation norm alone does not provide comparable concept access.This pattern also holds for Gemma, despite its larger norm variation than Llama and Qwen.
  • B Additional Directional-Encoding Results: Pointwise generation-token norm variation is relatively stable in instruction-tuned models, whereas the Llama base model shows elevated variation at later layers.This result is reported using the coefficient of variation across generation-token positions.
  • B Additional Directional-Encoding Results: Large radial variability does not imply norm-based concept encoding, because Gemma’s greater norm variation still accompanies the same directional-encoding pattern.The passage contrasts Gemma with Llama and Qwen models.
  • C CAA-m Per-Token Matching Algorithm: CAA-m chooses a separate additive coefficient for every token so that the normalized output reaches the requested concept score.Its construction uses the token norm, angular score, and orthogonal residual component relative to the concept direction.
  • C CAA-m Per-Token Matching Algorithm: CAA-m controls the angular concept score while allowing the hidden-state norm to change, separating these two geometric effects at the token level.The method’s per-token matching targets the normalized output rather than fixing the resulting norm.
  • C CAA-m Per-Token Matching Algorithm: The CAA-m expression is well-defined for γ ∈ (−1, 1), but the required additive coefficient can become large as |γ| approaches 1.This occurs when the residual component orthogonal to the concept direction is large.

D Additional Fixed-Angle Steering Results

At matched per-token target γ, S and CAA-m reach the same normalized concept direction but differ radially: S preserves the original norm, whereas CAA-m retains the additive norm change. At larger γ, CAA-m generally achieves lower perplexity and higher MMLU accuracy, while downstream task performance remains comparable on average but varies more.

  • Method comparison: At matched per-token target γ, S and CAA-m share the same normalized concept direction but differ in radial behavior.S preserves the original norm, while CAA-m leaves the additive norm change intact.
  • Per-dataset gaps: At low γ, S and CAA-m are close on all metrics.
  • Per-dataset gaps: At larger γ, CAA-m usually has much lower perplexity and higher MMLU accuracy than S.
  • Per-dataset gaps: At larger γ, downstream task metrics remain comparable on average but vary more across models and datasets.

E Additional Fixed-Strength Steering Results · F Concept-Score Closure · G Off-Arc Perturbations

Across fixed-strength steering, concept-score closure, and off-arc perturbations, token-level angular structure—not norm preservation alone—best explains stability and downstream behavior. Per-token targeting and the spherical arc produce tighter control or better outcomes because they preserve task-relevant angular organization.

  • E Additional Fixed-Strength Steering Results: Token-level angular profiles matter more than norm preservation alone in fixed-strength steering.CAA-r and AS preserve hidden-state norm, whereas CAA does not, yet the results emphasize angular structure as the stronger determinant.
  • E Additional Fixed-Strength Steering Results: At matched mean concept score, CAA-r and CAA have closely tracking downstream and PPL curves, showing post-hoc renormalization is not the main stability factor.The methods share the same normalized output direction after the additive update, while CAA-r only rescales the result to the original norm.
  • E Additional Fixed-Strength Steering Results: At high ¯γ, AS becomes substantially less stable than CAA-r despite both preserving ∥y∥= ∥x∥.CAA-r inherits token-dependent angular displacement from the additive update, whereas AS applies a fixed spherical displacement.
  • E Additional Fixed-Strength Steering Results: CAA-r requires model-dependent additive-strength calibration, whereas AS is less sensitive to activation norm scale because it is calibrated by angular displacement.The required additive strength depends on residual-stream scale, with Gemma requiring a much wider search range for α.
  • F Concept-Score Closure: S and CAA-m achieve near-zero per-token concept-score spread, while globally calibrated CAA, CAA-r, and AS produce much wider token-level ranges.The targeted methods explicitly solve for the requested score at each token; fixed-strength methods only match the target mean.
  • F Concept-Score Closure: Targeted methods create sharp achieved-score peaks, whereas fixed-strength methods yield broader distributions that diverge more at higher target scores.CAA-r and AS are calibrated to the same mean concept score and preserve hidden-state norm, but their distributions separate because their token-level angular profiles differ.
  • F Concept-Score Closure: Per-token targeted methods close concept scores more precisely, separating semantic strength from concept-score closure and occupying a distinct control–quality trade-off.Mean-matched fixed-strength methods express the target concept on average without applying the same intervention to every token.
  • G Off-Arc Perturbations: Moving away from the spherical arc worsens behavior despite fixed norm and concept score: PPL increases, while MMLU and downstream task metrics generally decrease.The degradation is approximately symmetric, strengthens with |δ|, and varies by direction type: random directions are mildest, PCA directions steepest, and cross-dataset directions intermediate.

H Additional Norm-Scaling Results · I Datasets and Data Sources · J Models and Licenses

The norm-scaling sweep separates semantic control from stability: fixing the angular edit makes β chiefly affect generation stability, especially under strong steering, while task metrics remain comparatively stable. The experiments use concept, auxiliary, and heterogeneous diagnostic datasets across Llama, Qwen2.5, and Gemma 2 checkpoints, with dataset and model licenses documented.

  • H Additional Norm-Scaling Results: H Additional Norm-Scaling Results: Fixing the angular component while multiplying norm by β leaves the target concept score unchanged and directly tests norm as an independent stability lever.β changes the steered activation’s radius, not its angular concept intervention.
  • H Additional Norm-Scaling Results: H Additional Norm-Scaling Results: At high γ, increasing the norm reduces the PPL penalty, whereas downstream task metrics vary little across β.This supports β as primarily a stability knob rather than a semantic-control knob.
  • H Additional Norm-Scaling Results: H Additional Norm-Scaling Results: The 70B model produces larger PPL ratios for strong angular edits, but larger norm scales still reduce PPL most strongly at high γ.The ordering over β remains unchanged in the larger model.
  • H Additional Norm-Scaling Results: H Additional Norm-Scaling Results: With angular edits fixed, norm has little effect on semantic task metrics but a large effect on generation stability, motivating γ as angular control and β as radial stability control.This summarizes the paper’s two-parameter view of steering.
  • I Datasets and Data Sources: I Datasets and Data Sources: Norm-variation diagnostics sample heterogeneous corpora spanning web, instruction, scientific, story, question-answering, toxicity, news, biomedical, and code domains.The mixture tests whether radial hidden-state geometry is stable across content domains rather than an artifact of one dataset.
  • J Models and Licenses: J Models and Licenses: Experiments evaluate Llama, Qwen2.5, and Gemma 2 checkpoints, while reporting dataset usage terms and model licenses for transparency without redistributing data or weights.Examples include Apache-2.0 for Qwen2.5-7B-Instruct, the Qwen Research License for Qwen2.5-3B-Instruct, Meta’s Llama Community License, and Google’s Gemma Terms of Use.
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